How does Ability Power affect the damage of skills?

I know that Ability Power increases the damage of the skill that your champion can use. However, I am not sure how it works. For example, does your skill's damage increase by 1 for every 1 point of Ability Power? If not, what is the rate of increasing damage per ability power? And does ability power affects skills that stuns or slows, or passive skills as well?

3 Answers

The ratio between the AP and the damage dealt (or special effects) is defined on a spell by spell basis. This wiki excerpt explains it better than I could:

For example, if a spell deals 100 (+20% AP) damage and a champion has 50 ability power, the spell's damage will increase by 10.
In-game, spell descriptions do not display the percentage, but do display the resulting bonus; the previous example would display as 100 (+10).

The spell description appears on muse over and extra damage is color coded:
1. Green - scales with AP
2. Orange - ... AD
3. Red - ... HP
4. Blue - ... Mana
5. Yellow - ... Armor
6. Magneta - ... Magic Resist

Some skills have things other than damage augmented by AP, like Janna's Zephyr.
The skill's tooltip should always tell you when this is happening. A good wiki or guide site could also tell you the ratio, that's obscured in the game for some reason I don't entirely understand.

Scaling by stats

Not all skills are increased by ability power, some are increased by other values like Health, Attack Damage, Armor and / or Magic Resist.

Here are 2 examples for different scaling:

Galio's Runic skin (passive)Galio gains bonus ability power equal to 50% of his total magic resistance

Lux's Illumination (passive)Lux's damaging spells mark any enemies with light energy for 6 seconds. Her autoattacks or Final Spark.png Final Spark will consume the mark, dealing 10 + (8 × Lux's level) (+ 20% AP) magic damage to the target.

Scaling ratio

The ratio for the specific skill differs per skill. All the abilities are unique, and therefore have their own scaling.

For example, let's take Lux:

Passive: IlluminationLux's damaging spells mark any enemies with light energy for 6 seconds. Her autoattacks or Final Spark.png Final Spark will consume the mark, dealing 10 + (8 × Lux's level) (+ 20% AP) magic damage to the target.

Q: IlluminationMAGIC DAMAGE: 60 / 110 / 160 / 210 / 260 (+ 70% AP) SECOND TARGET MAGIC DAMAGE: 30 / 55 / 80 / 105 / 130 (+ 35% AP)

W: Prismatic BarrierSHIELD STRENGTH: 80 / 105 / 130 / 155 / 180 (+ 35% AP)

E: Lucent SingularitySLOW: 20 / 24 / 28 / 32 / 36% MAGIC DAMAGE: 60 / 105 / 150 / 195 / 240 (+ 60% AP)

R: Final SparkMAGIC DAMAGE: 300 / 400 / 500 (+ 75% AP)

CC Scaling

Ability power (and other stats) may increase stuns, slows and passives, according on the skill. (however, this is not always the case and the scaling is, because the skills are unique, always different) For example:

Janna's Zephyr (W skill)PASSIVE: Janna is aided by a wind elemental, gaining movement speed and ignoring unit collision. MOVEMENT SPEED BONUS: 4 / 6 / 8 / 10 / 12% (+ 2% AP)

ACTIVE: Janna launches the elemental to deal magic damage and Slow icon slow an enemy's movement speed for 3 seconds (capped at 80% total). She loses the ability's passive benefits while it is on cooldown. MAGIC DAMAGE: 60 / 115 / 170 / 225 / 280 (+ 50% AP) SLOW: 24 / 28 / 32 / 36 / 40% (+ 6% AP)

Source: Lol Wikia

CC = Crowd Control (Slow, stun, fear)

I would like to add some useful tip, but I have no reputations yet to comment.

Upon mouse-over on the specific skill, you may notice that there are some texts/values that are in green, it means that the skill damage(or damage reduction, or speed increase, or whatever) scales with AP. Other skills does not increase damage even if AP is increased.

Here are some other details for the color scheme: