Armour refers to two different things: first, objects that you wear on your torso for protection (made of either leather or metal); and second, a number representing your total amount of protection against certain kinds of damage. We are primarily interested in this second definition.
The way armour is handled varies among the various beta releases. From beta 1 through beta 20, armour (less the attacker's armour penetration [APR]), was subtracted from the damage done by melee and archery attacks. For example, if a monster with 4 APR hits you for 20 damage, and you're wearing 15 points of armour, then you take 9 damage: the APR reduces your effective armour to 11, and 20 - 11 = 9. Under this scheme, armour is hugely effective in the early game, when most attacks are quite small. It does not work as well in the end game, against larger attacks.
From beta 21 through beta 24, a totally different scheme was used: the armour value determines a percent of damage reduction using the formula reduction = 1 - 0.99 ^ armour. (APR undergoes a similar function.) With typical early-game armour values, you absorb only a small percent of damage, which is rarely noticeable -- absorbing 10% of a 20-point monster attack means you still take 18 damage, vs. 10 damage under the older scheme. However, in the end game, with very large armour values, you absorb a considerable fraction of incoming damage. This forum thread describes the outcomes under various implementations and proposed implementations for the mathematically inclined.
From beta 25 onward, another totally different scheme is used -- armour provides a flat reduction (just like in betas 1-20) but only against a fraction of the damage. This fraction is controlled by your Armour Hardiness, a new number on the character sheet. A new talent called Armour Training boosts your Armour Hardiness (by 5% per talent point, up to 10 points, affected by mastery). The type of armour being worn also affects Armour Hardiness. Under this implementation, the early game becomes easier for heavy melee characters, but monsters are not completely shut down by high armour. On the other hand, investment in Armour Training will soak up generic talent points.