Every Hero has three unique abilities - one that can be used on enemies, one that can be used on allies and one that the hero can use on themselves. These are referred to in the wiki as the enemy ability, ally ability and self ability. This system means that there is no need to select which ability to use in combat - just click on a target and the appropriate ability will be used.
Enemies work in a similar manner, though they may lack one of the three abilities. Some bosses may also posses a special fourth ability that they use once per a fixed number of turns (unique to that particular enemy).
See Ability Power for exact information on how damage and other effects of abilities are calculated.