When I think mods, I typically also think PC gaming. But people do mod games on consoles, too—and it looks like modders are starting to sink their teeth into Destiny.
Right now, it looks like modders haven't figured out how to do anything particularly game-breaking yet. The main mod I've seen floating around is one that lets people shoot indefinitely, without ever having to reload or get more ammo. It's an infinite ammo/no reload glitch, as you can see above in the video by 0neThatsExiled.
Other people seem to be messing around with the mod, too:
Hopefully things don't get too out of hand with Destiny mods, given that it's a multiplayer game. We'll keep you updated with anything we learn.
EDIT: some people are getting worked up about the word 'mod' here. The people doing this are calling it mods, so that's what I called it, too. It can less disputable be described as a "cheat", and we've updated the headline to reflect that.