An Ability Counter is a specific type of Counter that can be placed on certain cards whose effect allows them to gain Ability Counters, with a usual maximum of 4. Such cards usually have an effect that allows them to gain Ability Counters during your Standby Phase, such as "Soul-Giver Priest". Other cards, have effects that allow them to place Ability Counters on other cards.
An Ability Counter itself does nothing. Certain cards require Ability Counters to use their effects, almost exclusively monsters. When a card with Ability Counters on it is flipped face down, all of its Ability Counters are removed.
For a list of cards involving Ability Counters, see List of cards involving Ability Counters.