I'm wondering if it is an increase to your hero's ability damage or if it is an increase to all units' spells (including heroes).
Ability power applies to things tagged Magicial, which is generally spells, with some exceptions. Conduits, for example, have a magical auto-attack.
Wondible's got it! :D
There's a bit more about the damage/defense systems on the FAQ, in the last section. Though I do think we can (eventually) do a much better job of explaining these systems through some in-game tooltips.