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Armor Penetration describes a character's ability to overcome the Armor Rating on an enemy to deal damage. Every point of Armor Penetration allows the character to ignore one point of armor for the purpose of calculating how much damage an attack deals. High armor penetration does not help a character to hit more often, it is purely used in resolving how much damage is done after the fact.
A character's Armor Penetration score is mainly comprised of the Armor Penetration rating on his weapon, although certain abilities gain a bonus to Armor Penetration when used.
There are several ways to increase Armor Penetration on your character:
A warrior with a longsword with +2 Armor Penetration attacks a Darkspawn with 4 Armor and hits for 8 points of damage. The Darkspawn's 4 Armor rating is reduced by 2 due to the warrior's Armor Penetration, resulting in total attack damage of: (8 Damage - (4 Armor - 2 Armor Penetration)) = 6 Total Damage.
Armor penetration is only used to mitigate enemy armor, it never increases damage itself.
Armor does not always provide full protection in all situations. The game models this by randomizing the Armor value on the target between 50% and 100% for each attack. For example, if the target had 20 Armor, the actual Armor value used for each attack would vary between 10 Armor and 20 Armor. Armor Penetration however is a constant without variance, making 1 point of AP more powerful than 1 point of armor.
Armor is resolved from the total damage calculated, including the critical hit multiplier. The only exception is bonus damage applied to constructs with the Shattering Blows talent and the effects of the Blood Frenzy talent, which are added after resolving the effects of Armor and AP.
Different weapon types have different armor penetration rating, and when comparing weapons, close attention should be paid to both its Damage and Armor Penetration ratings.