Quake 3 (and Quake IV) behave as described by ChrisF; armor protects you from a fixed 2/3 of incoming damage, assuming you have enough armor points to absorb the damage. (If you run out of health, you die, regardless of if you have any armor left or not.)
For the earlier games, the color of the armor you pick up matters too.
In Quake I and Quake II, green/Jacket armor provides 30% protection. Yellow/combat armor is 60%, and red/body armor is 80%. Armor shards from Quake II will increase your armor points, and the armor will be of whatever color you currently have; if you don't have ANY armor when you collect a shard, the points behave as if you had green armor.
The Energy Armor/Power Shield from Quake II provides 2/3 damage protection as well, but it's an item you switch on and off, and it uses your cell ammo instead of being a separate resource.