This achievement is unlocked by completing an optional, level 11 mission ("To Arms") that starts at the bounty board in Triton Flats. The quest seems to unlock after finishing
This mission requires that you donate a total of 50 common (white) weapons* to a Troll Booth located also in Triton Flats.
* This quest will not accept Oz kits / grenade mods / class mods / shields. It must be a (white) weapon. Also, it has been reported that you can trade in quest weapons, so be careful that you don't need to reset a quest because you turned in a required weapon. (Zapped.)
Every 10 items you donate, the quest will update and request that you do some random chore (ie: retrieve stolen guns, defend the booth, etc). Once you donate 50 items, you complete the quest by returning to Concordia; curtains are pulled to reveal a
...and then you finish the quest by turning it back into the bounty board. The achievement should pop upon doing so.
There are a few different ways to obtain common weapons, (and these techniques shouldn't surprise anyone:) either through mob drops, winning pulls on slot machines, or exploits within the game.
Personally, I found the bulk of my white weapons through the slot machines in Moxxie's Up Over in Concordia. Between myself and a friend, we were pretty quickly able to pool together enough white weapons to finish the quest.
Happy sailing and best of luck. The achievement should unlock after completing the quest by turning it into the bounty board.
A helpful tip: If you haven't been spending Moonstones and have quite a few stored, it might be a good idea to start unlocking the backpack upgrades. I did that first; as having extra slots to hold white weapons made things go a little faster.