Currently Smoking: Whatever the generic is from 7-Eleven

The Dungeons & Dragons Player's Handbook is the character creation guide for what Wizards of the Coast admits is the fifth edition of "the world's greatest roleplaying game".  As of this writing, the Monster Manual for this edition is just out but the Dungeon Master's Guide is not; presumably those who want to start a new campaign will have to use the tools presented in modules and other materials like The Lost Mine of Phandelver.  But of these, the Player's Handbook is obviously the "core" of the corebooks that would give us all an idea of how this is all supposed to work.

The Book:

I am not a big fan of the cover, or the general art design which is shared with the Monster Manual cover and the recent modules.  The painting style (in this case a battle with a fire giant king) seems rather dark and blurry.  And as one of my friends pointed out, the strip down the back cover which contains the promo text is of a rougher quality than the gloss on the rest of the cover, and is more likely to wear quickly.
The interior artwork is generally good but uneven, in my opinion.  My favorite part is the cartoons in the Appendix showing examples of character conditions like "frightened," "prone" and "unconscious."

The Introduction goes over not only the general concept of a roleplaying game but specific concepts dating back from at least the development of the d20 System (like, when you roll on d20, you always want to roll high), along with elements new to this edition.  As with 4th Edition, two core principles are "specific beats general" (e.g., if you have an ability that allows a certain action, you can have that option even if it's not allowed for other characters) and "round down" (even if the fraction is half or above).  There's also a new core element: Advantage vs. Disadvantage.  Simply, if you have advantage on a d20 roll, you get to roll the d20 twice and take the better result.  Whereas if you have disadvantage (say, a Drow Elf trying to attack someone in daylight) you have to roll d20 twice and take the worse result.  Advantage and Disadvantage replace a LOT of the little modifiers that were used to distinguish in-game conditions, leading to an overall streamlining and "cleaning up" of the mechanics.

Part 1: Creating A Character

This is the meat of the book.  It starts with Chapter 1, a step-by-step guide to creating your character, where you first choose a race (defined in Chapter 2) then a class (Chapter 3) and then generating ability scores.  There are three ways to do that: roll best 3 of 4d6 for each stat, use a set array of stats ranging from 8 to 15, and a point buy system that also will not allow you to get a starting score higher than 15.  (My friend who is starting a new D&D campaign let us roll the 4d6 stats and after seeing most of us get most of our stats at 16 or more, decided he's going to stick to the point-buy system in the future.)  As in other d20 games, 10 is an average and every 2 points above 10 is a +1 with that ability (and every 2 below 11 is -1).  The book explicitly states several times that even with racial adjustments, level-based adjustments and feats, an ability score cannot go above 20.
From there you describe your character, using his race/class background and traits that suggest themselves from having a high (or low) ability score.  Then you get your equipment (described in Chapter 5; you can get a lump sum of gold pieces to buy stuff as in other editions, but here the default assumption seems to be that you get set packages of gear for both class and background, similar to the 'outfits' in Legend of the Five Rings).  Then you "come together" - just as you have to describe your own character, you and the other players should figure out why you're together.
Chapter 1 also touches on earning experience; as in other d20 editions, every class uses the same XP chart, which starts rather low (only 300 XP to reach 2nd level).  
This chapter also details another specific core concept of 5th Edition: The proficiency bonus.  All characters have the same level of proficiency bonus (+2) at 1st level, going up generally +1 every additional 4 levels, and this is based on character level, not total class levels, so a multiclassed 2nd level Fighter/5th level Wizard still has only a +3 bonus, not +5 (2 plus 3).  Proficiency bonus applies to what previous editions applied to Base Attack Bonus, saving throw bonuses, and skill points.  As with prior editions, the difference between classes is largely where the bonuses go, but in this case it's all one mechanic.  For instance a Fighter applies the bonus to attacks with all weapons and has "proficiency" with saving throws based on Strength and Constitution, meaning he applies the proficiency bonus to those saves in addition to the ability modifier.  A Rogue applies the bonus to his skills (as do other characters, although they don't get the same skills).  A Wizard or other spellcaster applies the bonus to both attack rolls with his spells and to a base DC of 8 used to save against those spells.  As with advantage/disadvantage, this concept does a lot to define the character abilities in the following two chapters.

The race chapter (Chapter 2) is noteworthy for at least two things: One, the fact that certain races (Dragonborn, Gnomes, the half-humans and Tieflings) are listed out of order from Dwarves, Elves, Halflings and Humans as being optional.  This seems to be part of a "modular" approach in which the game allows the DM to pick and choose what elements will be used instead of assuming them all as required (for that matter, it's also stated that Drow is an allowable subrace of Elf, but you should check with the DM to see if you can play that race).  The other noteworthy thing is how they introduce each race with character fiction mainly from Forgotten Realms books (the sample character in Chapter 1 is Breunor the Dwarf, and yes, the sample Elf in this book is Drizzt).  Later, they use both Drizzt's nemesis Artemis Entreri and the Dragonlance character Tika as examples of how to develop character background from completely different perspectives (good-hearted barmaid vs. cold-hearted mercenary).  It's a subtle and effective way to employ the rather vast amount of lore that this game has built up under various owners, and it inadvertently points out that D&D is a Fantasy genre unto itself.
Otherwise, the race section gives more detail on how the system works.  Racial abilities are described in terms of the above rules: Elven senses are defined as proficiency with the Perception skill, while Dwarven poison resistance is an advantage to saves.  Notably, all races including Humans get ability score increases.  But as part of the modular approach, with Humans you can either have +1 to all stats or +1 to two stats along with an extra skill and a feat, which you normally can't get at first level and is one of those optional elements the DM doesn't have to use.  (See below)

Chapter 3 has twelve classes: Barbarian, Bard, Cleric, Druid, Fighter, Monk, Paladin, Ranger, Rogue, Sorcerer, Warlock and Wizard.  In some respects they all follow the same pattern.  Again, all classes have the same proficiency bonus at the same character level, but the bonus applies to different things.  Almost all classes reach a point at 3rd level where they have to follow a certain "archetype" - for instance Rangers have to choose between Hunter (more combat ability) vs. Beast Master (having a pet who can fight in combat).  Two of these archetypes, the Eldritch Knight for Fighter, and the Arcane Trickster for Rogue, are very closely adapted from 3E prestige classes, and effectively make the character multiclassed with spellcaster abilities.

What's really different about the new edition is the spellcasting system.  It is old-style "Vancian" fire-and-forget spellcasting.  Sorta.  However even though many of the classes have a limited set of spells known and most of them have to prepare their spells in advance, there is no relation to which spells you memorize versus the number of spell slots you have.  That is, if a 3rd level Wizard has four 1st level spell slots and two 2nd level slots, he COULD memorize four different 1st level spells and two 2nd level ones to cast once per day each.  OR, he could memorize Magic Missile and cast it four times.  He could even take one of his unused 2nd level spell slots to cast Magic Missile, which boosts the spell effect (see Part 3, below).  Also, not only do spellcasters use different ability scores for their spellcasting rolls, their spellcasting doesn't all work the same way.  Clerics, Druids and Paladins can choose to prepare any spell from the lists for their classes, but Rangers (who are also 'divine' casters) have a set list of spells known, but can cast them "spontaneously."

Chapter 4 details the background elements of your character.  This is where it's mentioned that your character "doesn't have to conform to binary notions of sex and gender."  There is a random height/weight chart for each race, similar to the Pathfinder table.  There are alignments, nine of them this time.  And in this edition, you are allowed, or obliged, to make up a background for your character, with the backgrounds in this chapter mentioned as samples (it's mentioned that the example characters don't conform entirely to these rules, and players should be allowed to modify them as they see fit).  But this is still one of the more fun sections of the book, in terms of the flavor text they give to describe a person's origins, motivation, 'bonds' and flaws ('In fact, the world does revolve around me').
A primary reason to pick a background is that it applies addditional equipment, two other skills and possibly other proficiencies to your character; for instance the Noble has both History and Persuasion skill, along with skill in one language.  Some of the backgrounds are natural fits to classes, such that in Chapter 3, they recommend them for 'quick builds' (Soldier to Fighter, Sage to Wizard, etc.).  Another reason is that the background serves to justify the award of inspiration, similar to how the GM in a point-buy game can use a PC's drawbacks against him or award "hero points" for role-playing his background and motivations.  In 5th Edition, however, this concept is as basic as everything else: If the DM awards inspiration, it means you get to apply advantage to a future roll of your choice, and you can't "bank" multiple inspirations for later.

Chapter 5 is the equipment chapter.  If you're not using the equipment given to your character for his class and background from the other two chapters, you use the random-roll sum of funds they give you here.  Compared to even 4th Edition, this section is really stripped down.  The weapons section is only one page.  Armor thicker than light has limited (or no) ability to add your Dex bonus to AC.  Shields are only shields.  Not bucklers, not tower shields, shields.  They give you +2 AC.  Period.
This section does also include rules for "lifestyle", similar to those in Pathfinder or Shadowrun, in which 1 GP a day will provide a "modest" default standard of living.  There is also a d100 table that a character can roll to get one of an extremely random range of trinkets.

Chapter 6 deals with "customization options" that are specifically listed as optional: multiclassing and feats.  Fifth Edition seems to strike a middle ground between the two prior editions with regard to multiclassing.  You still have to advance one class level at a time (as opposed to the AD&D method of taking multiple classes and dividing the XP or the 4th Edition method of swapping out feats and powers from another class while actually only having your original class).  But you normally have to have an ability score of 13 in the prime requisite(s) of your new class, on the premise that joining a class that you didn't originally train for requires being a "quick study".  You also don't get all the proficiencies that the new class would have given you if you'd started it at 1st level.  And if both of your classes are spellcasters, you get the spells you learned from each class individually, but the spell slots you can actually use per day are averaged out on a special table.  Whereas Pathfinder seems almost to require multiclassing, this version of D&D seems to discourage it.  Not nearly so much as D&D 4, but with the proficiency system, clearly there isn't much need for it.  For one thing the character creation rules play to certain tropes; if you want to be an Elf who can use a sword and cast spells, it's possible to be a single-classed Wizard with a sword because High Elves get proficiency with swords and bows.
The next section is the feats.  In standard rules, characters are supposed to raise attributes by 2 points (+2 to one stat or +1 to two) at 4th level and every four levels thereafter (except your final boost is 19th level for some reason).  To take the feat option (IF the DM allows it) you have to sacrifice your ability increase for a certain level.  However a lot of these feats train the character in such a way that one of his stats is raised by 1 along with another benefit (for example the Observant feat allows you to increase Intelligence or Wisdom by 1 and also get the ability to read lips and a +5 to passive Perception and Investigation skills).  Of course this means you can only get a maximum of five feats by 19th level (six for a Human) so there's only so many to choose from compared to 3rd Edition iterations.  But they also seem a lot more potent.  

Part 2: Playing the Game

This section begins with Chapter 7, in which the basic game mechanics are elaborated on.  For instance, if you have effects that create both advantage and disadvantage, they cancel out, even if  you have more advantage than disadvantage conditions or vice versa.  The chapter centers on the proficiency bonus in combination with the ability check: Sometimes the proficiency bonus is halved or even doubled (for instance the Rogue's Expertise feature allows one to choose two skills or one skill and thieves' tools proficiency and double the proficiency bonus with them).  The proficiency bonus of course adds on to rolls made with character abilities, which are gone over here.  This means that Chapter 7 is basically the skills system, such as it is.  There are only 18 skills in the game, and most characters will be lucky to have six.  Anything that isn't specifically mentioned as a skill can be done as a raw ability check, sometimes modified by a character's proficiency with certain tools.  For instance, there is no Disguise skill in the game.  However, under Intelligence it's mentioned that one use of the ability is putting together a disguise to pass as a city guard.  Certain backgrounds (namely Charlatan) give a character proficiency with a disguise kit, meaning that if such a character uses that kit, he can add his proficiency bonus to Intelligence checks for disguise.

Chapter 8 (Adventuring) deals with a great deal of in-game and downtime material, such as terrain, marching, lighting conditions, and what happens if you go too long without food or water .  In that case, you get the condition of "exhaustion," further detailed in Appendix A, in which a character suffers increasing levels of disadvantage and ability loss, potentially leading to death.
Most notably this chapter includes the rules for resting and healing.  A "short rest" is a downtime period of about 1 hour.  During this time a damaged character can roll his actual Hit Dice- equal to what he rolled for hit points (a 4th level Fighter gets 4d10 for Hit Dice) -to see what he regains for resting.  A "long rest" is a period of at least 8 hours (although they mention that a character can stand watch up to two of these hours) with no more than one long rest per 24-hour period.  After this period  a character regains all damage.  This is probably the rule most similar to 4th Edition, and it basically means that "resource management" is a per-day affair.  Many character abilities (such as your number of spell slots) recharge after a long (sometimes short) rest.

Chapter 9 is the combat chapter.  Most of this should be familiar from other d20 games.  Some rules are specific to this edition and are fairly easy in execution although not necessarily explained clearly.  For instance, many characters have what are described as "bonus actions."  It's only here that they clarify that bonus actions only apply with class abilities described as such, and otherwise a character doesn't have a bonus action.  Also, you can't take more than one bonus action per turn, even if you have more than one such ability.  There are also actions described as "reactions" that allow you to interrupt another character (the infamous opportunity attack is an example of a reaction).  Other actions in this chapter use the current set of mechanics; for instance if you use the Help action to assist another character it either gives them advantage on their next ability check or distracts an enemy in combat allowing your ally to gain advantage on the next attack roll.
Attack rolls involve your relevant modifier and any proficiency bonus.  It's important to note that, as with Palladium, NOT having proficiency with a weapon doesn't mean you get a "non-proficiency penalty" with it, it just means you don't get the proficiency bonus.
When a character is knocked to 0 hit points or less, it is instantly killed if it takes damage in excess of its negative maximum hit points (a character with 12 hit points is instantly killed if the damage that took her to 0 also brings her to -12).  Otherwise a character at 0 or negative has to make death saving throws, with no ability modifier, trying to reach 10 or more.  If you make three of these saves, you stabilize and don't bleed out.  If you fail three, you die.  This is also take n directly from 4th Edition.  They do give an attacker the option to render an opponent unconscious instead of attempting to kill him; when this action is declared, the "killing" shot instead makes the target both unconscious and stable.

Part 3: The Rules of Magic

This section is dominated by Chapter 11, which is the actual list of spells.  But before this, Chapter 10 details the rules of spellcasting.  In addition to the aforementioned variable spell slot system, some spells are listed as 'ritual.'  This means that if a character is a ritual caster (either because of his class or a feat) and he knows a ritual spell, he can cast it without using a spell slot, if he takes 10 times as long as usual.  This also means that you can't "boost" the effect with a higher level spell slot.  Certain spells in the Chapter 11 list have a paragraph marked "At Higher Levels" showing what happens if you cast the spell with a slot higher than its level.  For instance if you cast the 1st level Charm Person with a spell slot above 1st you can charm one additional person within 30 feet for each spell slot level above 1st (e.g. three people with a 3rd level slot).
The actual spells are often quite powerful (even attack cantrips can do 1d6 or 1d8), and in some cases (like Feather Fall) can be cast as a reaction.  As for major effects, using Resurrection on a being dead more than a year means you are at disadvantage on all rolls and can't cast spells again  until completing a long rest, and if you use Wish for other than a set list of effects, not only are you debilitated, there's a 33% chance you will never be able to cast Wish again.

Appendices

In Appendix A, you have the aforementioned section that shows not only examples of game conditions but what they mean in rules terms.

Appendix B is a section showing the names and main portfolios of various pantheons of deities, from not only D&D's proprietary lines like Greyhawk and Forgotten Realms but also historical pantheons like the Greek and Norse.  While D&D veterans will have much more background material on deities from previous books, this section is still important in that it shows the Domains the gods grant their clerics in 5th Edition.

Appendix C details The Planes of Existence and the cosmology of the D&D universe.  It very much resembles 4th Edition, including both the Feywild and the Shadowfell, which in terms of previous cosmology are defined as "material echoes" of the Prime Material, touching upon the Positive and Negative planes respectively.  The center of the Outer Planes arrrangement is Sigil.  And there is a Far Realm (since Warlocks are allowed to pact with 'Outer' beings).

Appendix D is a short list of basic monsters for low-level games, including the animals PCs might interact with.  

Appendix E is a list of recommended reading for Fantasy (including both George RR Martin and China Mieville).  And then there's an index and character sheet template.

SUMMARY

Wizards of the Coast's stated goal with the new D&D was to have something that was in some way compatible with all the previous editions.  Remarkably, the new Player's Handbook does a lot towards achieving that goal.  It has much of the streamlining of both 3rd and 4th editions but avoids both the "3.5 turned up to 11" approach of Pathfinder and the "player character as chess piece" mechanics of 4th Edition.  Doing so, it greatly resembles Gygaxian AD&D, even though that was in some ways more complex.

I still prefer Pathfinder because that has more options, but that isn't what this game emphasizes.  It's easier to teach, easier to make a character, and easier to get things rolling.  It certainly seems like it'd be easier and faster to run a combat than in D&D 4, but how could it NOT be?

It's been mentioned by many others that whether you like Dungeons & Dragons or not, it is the flagship of the industry, and generally its success (or failure) will be that of the gaming industry as a whole.  It's thus appropriate that the new version of the game is both well-produced and user-friendly while still keeping all the stuff that makes D&D "D&D."  So far, with its introductory material, Wizards of the Coast looks to have set Dungeons & Dragons on a bright new future.