Ability damage is one of the game's archetypal damage types, and is the damage type dealt by most abilities. Ability damage is the pre-requisite for applying ability effects, such as 
Spell Vamp and 
Rylai's Crystal Scepter. Dealing ability damage will also generate aggression from turrets but, by contrast to basic attack damage, does not trigger the Call For Help AI on minions.
Ability damage is never described as such in-game, instead being listed as physical, magic or true damage. However, those types are attributes that can apply to other archetypal damage types (e.g. basic attack damage) - and so is not relevant.
Non-Damaging Abilities
The following non-damaging effects actually deal 1 ability damage, inflicting as true damage:
Pillar of Ice
Time Bomb- This list may be incomplete.
Non-Abilities
The following non-abilities will also trigger spell effects:
- Bilgewater Cutlass
- Blade of the Ruined King
- Chilling Smite
- Hextech Gunblade
- Hextech Protobelt-01
- Hextech GLP-800
- Luden's Echo
- Runic Echoes
- This list may be incomplete.
On-Hit Ability Damage
The following on-hit effects deal ability damage, meaning they will trigger ability effects:
- Ekko's Phase Dive
- Gragas' Drunken Rage
- Illaoi's Harsh Lesson
- Jax's Empower
- Grandmaster's Might (passive)
- Kassadin's Nether Blade (active)
- Kayle's Righteous Fury
- Kennen's Electrical Surge (passive)
- Kha'Zix's Unseen Threat
- Leona's Shield of Daybreak
- Mordekaiser's Mace of Spades
- Twisted Fate's Stacked Deck
- Udyr's Tiger Stance (activation effect)
- This list may be incomplete.
Attack Effects
As of V7.22, the following abilities now apply both attack effects and spell effects.
Note that, however, the damage dealt by these abilities is not classified as ability damage, except in the case of 
Whiplash and the magic damage portion of 
Urchin Strike. Except in those cases, the damage dealt by these abilities will not benefit from Spell Vamp. It is possible—as with Non-Damaging Abilities (above)—that these abilities actually deal 1 ability damage.
Steel Tempest is currently bugged and is only occasionally triggering spell effects.