Armor penetration

First note that there is a big difference between "penetration" and "reduction". Players usually just call all of the effects "penetration" but there is actually a difference between the two, as far as the game code is concerned. Armor penetration is a champion statistic that allows the champion to ignore some or all of a target's armor. All champions have 0 base armor penetration but can increase it with items, runes, masteries and some champion abilities. This cannot affect enemy armor beyond 0 to deal bonus damage.

Armor reduction is the effect of some items and champion abilities that reduces the armor of one or more targets for a few seconds. This allows allies to benefit from the reduction of armor. This can reduce below 0, which means bonus damage.

Calculations
Next, note that when attacking, armor penetration and armor reduction are applied to the target champion's armor in the following order:
 * 1) Percentage armor reduction
 * 2) Flat armor reduction
 * 3) Flat armor penetration
 * 4) Percentage armor penetration

Percentage armor reduction
Percentage armor reduction stacks multiplicatively.
 * The target's armor is reduced by a percentage.
 * If the target has 30 armor and you have 20% armor reduction, the target will take damage as if it had only 24 armor.
 * 30(300.2)30624
 * Percentage armor reduction has a bigger impact on targets with higher resistance.
 * For 40% reduction, a target with 200 armor will lose 80.
 * For 40% reduction, a target with 40 armor will lose 16.

Flat armor reduction
Flat armor reduction stacks additively.
 * The target's armor is reduced by a certain amount.
 * If the target has 30 armor and you have 20 armor reduction, the target will take damage as if it had only 10 armor.
 * 302010
 * Armor reduction can reduce a target's resistance below zero. For example if an enemy only has 40 armor, and you have a Black Cleaver, which removes up to 45, then they will be at -5 and take bonus damage.

Flat armor penetration
Flat armor penetration stacks additively.
 * The target's armor is ignored by the amount of the attacking champion's armor penetration.
 * If the target has 30 armor and you have 10 armor penetration, the target will take damage as if it had only 20 armor.
 * 301020
 * Flat armor penetration cannot reduce a target's armor below zero.

Percentage armor penetration
Percentage armor penetration only exists in three instances, being the item Last Whisper item, the Weapon Expertise talent, and the Crystal Scar aura. These effects stack multiplicatively, which means a max of 46% in Classic or 53% in Dominion.
 * A percentage of the target's armor is ignored by the amount of the attacking champion's armor penetration.
 * If the target has 30 armor and you have 20% armor penetration, the target will take damage as if it had only 80% of that. So if the above effects reduced a targets armor to 30, or they just have 30 and you have no other above effects except 20% here, then it would become:
 * 30 x .8 = 24, meaning you hit them as if they had 24 armor instead of 30.
 * Percentage armor penetration has a bigger impact on targets with higher resistance. Assuming you have the last whisper without the talent:
 * For 40% penetration, a target with 200 armor will lose 80.(200 x .6 = 120)
 * For 40% penetration, a target with 40 armor will lose 16.(40 x .6 = 24)
 * Pecentage armor penetration cannot reduce a target's resistance below zero.

Examples
You have 10 flat armor penetration, and 40% armor penetration. And the targets are affected by 30% armor reduction and 20 flat armor reduction,
 * Target A has 80 armor.
 * The 80 is reduced to 56 by the 30% armor reduction.
 * The 56 is reduced to 36 by the 20 armor reduction.
 * The 36 is reduced to 26 by your 10 armor penetration.
 * The 26 is reduced to 15.6 by your 40% armor penetration.
 * 26(260.40)2610.415.6
 * Target A takes damage as if it has 15.6 armor.
 * Target B has 18 armor.
 * The 18 is reduced to 12.6 by your 30% armor reduction.
 * The 12.6 is reduced to &minus;7.4 by your 20 armor reduction.
 * The &minus;7.4 is not reduced by your 10 armor penetration (cannot penetrate below 0).
 * The &minus;7.4 is not reduced by your 40% armor penetration (cannot penetrate below 0).
 * Target B takes damage as if it has &minus;7.4 armor.

Negative armor
Armor "Penetration" cannot reduce a unit's armor below 0, but armor "reduction" can. This is unlikely however because the "reduction" effects are applied first. However, abilities that cause flat armor reduction (such as or ) can still reduce a unit's armor below 0 too(They are reduction, not penetration). If a target has negative armor, the amount of bonus damage is equal to the amount of damage reduction caused by that positive amount of armor, meaning you receive a bonus in damage over 100%. See below:
 * If a target has 10 armor, they take 9.1% reduced damage (formula from Armor).
 * Therefore a target with &minus;10 armor would take 9.1% bonus damage.
 * So if you attack for 100 physical damage on the target, it would take 109.1 damage instead.
 * If a target has 20 armor, they would take 16.7% reduced damage.
 * Therefore a target with &minus;20 armor would take 16.7% bonus damage.
 * So if you attack for 250 physical damage on the target, it would take 291.7 damage instead.

Champion abilities
Note: Only the armor penetration effect of these abilities is shown here, to read more information on each of these abilities, follow the link on each of them.
 * passive grants him 10 / 20 / 30 armor penetration.

Masteries

 * : Gives 10% Armor Penetration.
 * : Gives 2 armor penetration (6 at rank 3)

Runes

 * runes provide flat armor penetration. These are commonly purchased because, especially along with armor penetration quintessences and against an enemy with low starting armor, armor can be brought very low, if not to zero, from the very beginning of the game and armor penetration is hard to acquire otherwise from items or abilities.

Champion abilities
Note: Only the armor reduction effect of these abilities is shown here, to read more information on each of these abilities, follow the link on each of them.
 * fires automatically every half second, reducing enemies armor with each hit by 1 / 2 / 3 / 4 / 5 for 2 seconds.
 * reduces its target armor by 10 / 14 / 18 / 22 / 26 for 5 seconds.
 * , while in ranks 2, 3, 4 and 5, the turrets' attacks reduce armor and magic resist by 1 per hit stacking up to 50 times.
 * reduces the armor of all enemies hit by 10 / 14 / 18 / 22 / 26 % for 3 seconds.
 * reduces the armor of all enemies that walk through the wall by 15 / 20 / 25 / 30 / 35 for 5 seconds.
 * reduces the armor of enemies hit by her autoattacks by 3% for 5 seconds. This debuff stacks up to 5 times.
 * reduces the target's armor by 5 / 10 / 15 / 20 / 25 for 4 seconds.
 * reduces the armor of all enemies in the area by 20 / 25 / 30 / 35 / 40 for the time they are on it plus 1 second.
 * reduces the armor of enemies that step on the trap by 20 / 25 / 30 / 35 / 40 % for 12 seconds.
 * reduces the target's armor by 10 / 15 / 20 / 25 / 30 for 1 / 1.5 / 2 / 2.5 / 3 seconds.
 * , when empowered by having more than 50 Fury, Dice will reduce the armor of all enemies hit by 15 / 17.5 / 20 / 22.5 / 25 % for 4 seconds.
 * reduces the armor of all targets hit by 15% for 4 seconds.
 * reduces the armor of all enemies hit by its activation damage by 10 / 15 / 20 / 25 / 30 for 4 seconds.
 * targets an enemy to steal their armor by 15 / 20 / 25%, this amount is doubled over the next 6 seconds.
 * reduced the armor of all enemies hit by 12 / 14 / 16 / 18 / 20 % for 5 seconds.
 * reduces the armor of the target hit by 30% for 3 seconds.

Others

 * While playing on the Dominion game mode, every champion gains 12% armor penetration.