Relics are items that provide additional random abilities, and are unlocked after the following prerequisites have occurred:
1) You have ascended at least once (See the World Traveler achievement)
2) After ascending, you have reached the 99th Zone (Bloodlands Lvl 99), right before Omeet at Zone 100
3) You find and kill the 'Relic Ooze' monster
Per the community wiki page for Relics, the Relic Ooze will be the very first monster of a Zone if it will appear at all in that Zone. In addition, there is a range of zones it can appear at, depending on your starting zone for a new ascension and the highest zone you've ever been in all your previous ascensions. Generally, this should only affect you if you've progressed past Zone 150 in your very first run without ever ascending. Otherwise, Zone 99 should have your very first relic.
Upon receiving your first relic, you will unlock this achievement.
Meanwhile, after getting your relic and achievement, several more things will occur:
1) A new tab (a chest icon) will appear on the right side of the screen, holding information on relics and forge cores
2) A new store option will appear, selling three relics for 40 Rubies
3) No more Relic Oozes will appear in the current Ascension
To earn more relics, you can either pay the 40 Rubies for a set of three, or perform another Ascension and work your way back up to Zone/Lvl 99 to find another Relic Ooze monster.