The abbreviation "AP" means "Ability Power" in League Of Legends. AP is one of the two main damage types in LOL, along with AD, which is Attack Damage. They are stats like Strength or Intelligence in other games that can be improved by items or runes.
Champions that scale via AD do strong, continuous damage with physical attacks with their basic attacks. Other champions, on the other hand, scale via AP because they rely primarily on magical damage. Such champions are often found in the mid lane.
The AP value of such champions influences how strong certain abilities are and how much damage they do over the course of a match.
The distinction between AD and AP usually refers to the build of a champion. These can be played on AD or AP depending on suitability, preference and strategy and thus differ greatly from each other.