It is an ability, just to clarify. It is an ability that is harder to get and generally better than the other ability/abilities.
They are in every main Pokemon game from Gen V onwards (not sure about spin-off games) and just do something different to a Pokemon.
For example: Talonflame's normal ability is Flame Body. If you make contact with it, it has a 30% chance to burn you.
But Talonflame can also have the ability Gale Wings. This ability will give all of Talonflame's flying moves priority. Priority means that they go ahead of other moves.
There are different ways to get Hidden Abilities in every game. In X/Y (where talonflame is found) you can find a HA Fletchinder in the friend safari. You evolve it to talonflame and you have a HA Talonflame.
I think this covers it, but if you have any other questions please ask in comments!