There are 15 Relics, all obtained the same way. They spawn deep underground and you'll hear a weird high-pitched noise when your near one that becomes more intense as you get closer. As an FYI, they'll actually spawn floating in the air if you are on a Low World Type. Once unlocked, you'll keep the Relics even after deleting the World.

Load into a World and hit F1 on your keyboard. This will activate Wireframe Mode, making all the blocks transparent. You know what to do from here.
Simply walk around and dig down to them using the laser or pickaxe.

Go to the Main Menu and click "View Weapons". It will list every Relic (as well as every tool) and what they do.