In order to redeem these codes, you first need to reach main town of Concordia. You'll be able to visit Concordia after a few opening missions in the campaign. Here you'll find a Shift Code Kiosk where you can input these codes. Alternately, you can enter them through the Options menu.
Most SHiFT codes will award you one or more Golden Keys, which you can also pick up at the Kiosk. The keys allow you to unlock the loot chest to the north, near Moxxi's bar and Springs' shop. Each time you open the chest with a Key, you'll get up to four weapons or shields of purple quality. They'll be either your character's level or one level below.