Weapon Tiers were added in build v20.0 (2/22/13). Formerly, they would allow the player to improve their weapons by spending stars (which were earned through completing contracts) on a weapon. Every slot has 3 tiers, the first being unlocked by default. Higher tier parts are more specialized and provide a higher bonus than the lower tier parts.
As of build v39.0, (11/25/13) contracts were removed from the game. However, weapon upgrades are now made available by using a weapon part to gain a certain amount of xp on it, then spending Blutes to upgrade said part to the next tier.
Tiers, and Upgrade Points were introduced into Loadout to give players, something to stride for by allowing them to improve their favorite weapons a little bit without changing them enough to be game breaking. Upgrading weapons with parts from the higher tiers only will generally increase the gun's overall performance by 10 or so percent depending on the build and upgrades chosen.
Video detailing the former weapon upgrade system: