Recently in a gaming session the player were in a ship chasing another ship that was slightly faster than the player´s ship.
The players' ship had no experience captain, but they had a "good navigator". The ship they were chasing had an experienced captain and navigator.

We tried to find something on the topic in the core books and the only thing we found was a brief and general description of some types of boats on the Dungeons Master Guide, if I´m not mistaken. There is no skill related to that, and there was none in 3.5 as well. This is very strange because if you think about it, up until very recently, transport through the seas was by far the fastest way to travel, and should be fairly common.

Since we did not even know what skill to use, I just described what happened, that the players lost sight of the boat but eventually found it again by following the coast. But just is just telling a story ain't that fun, and besides, what if

a)   The players had to travel during a storm?
b)   The players had to go through deep seas?
c)   What if I wanted to have opposing skill checks to see if they were skilled enough to reach the ship they were chasing?

Tool Proficiencies: Navigator’s Tools and Vehicles (Water) seem to be what you are looking for.

QuoteNavigator’s Tools. This set of instruments is used for navigation at sea. Proficiency with navigator’s tools lets you chart a ship’s course and follow navigation charts. In addition, these tools allow you to add your proficiency bonus to any ability check you make to avoid getting lost at sea.

QuoteVehicle Proficiency. If you have proficiency with a certain kind of vehicle (land or water), you can add your proficiency bonus to any check you make to control that kind of vehicle in difficult circumstances.

Other skills could be usable as well. Perception may be needed to sight a ship in a storm and Athletics to stay upright on a deck. But the above are essentially skills when using those tools.

EDIT: And to support my comments, I note the Sailor Background gets the Skill Proficiencies - Perception and Athletics and Tool Proficiencies - Navigator’s tools, vehicles (water)

actually 3.5 did have vehicle rules there hidden away in the arms and equipment guide.

I don't know 5e, but I do know sailing, intimately.  If the party found the ship again on the coast, it sounds like they were coast-hopping.

In which case, this went exactly wrong.  If you're coast-hopping, a navigator is superfluous, and if the other side has an experienced shiphandler and you don't, and they have a faster ship than you do, then there are only three ways you're not completely fucked: if the other crew's a bunch of incompetents, if enemy action delayed the other ship, or sheer dumb luck and lots of it.

The reason a navigator (and, thus, navigation tools) is superfluous is obvious: you're following the coast, right?  There's the coast: just avoid running aground, something for which you want a shiphandler, not a navigator.

makes sense

Quote from: tuypo1;817801makes sense

Miracles sometimes occur.

Quote from: Skywalker;817783Tool Proficiencies: Navigator's Tools and Vehicles (Water) seem to be what you are looking for.

And what ability score would be appropriate for this type of test?

Quote from: bla4ck;817973And what ability score would be appropriate for this type of test?

Depending on what you are using the Tools far, as per usual.

For Navigator’s Tools:
- to chart a course on a map, Int
- to use stars and geographical features, Wis

For Vehicle (Water)
- steering a course through rocky shoals, Wis
- holding a sail rigging together in a storm, Str
- avoiding being rammed, Dex
- chasing down an enemy ship over an extended period of time, Con

I'd also grant Advantage or Disadvantage on handling a ship if the wind and tide are particularly favourable or unfavourable.

Fortunately, for more guidelines you can always refer to earlier edition rulebooks.