Basically you signed up at redditgifts.com and linked your reddit account to it. Then when you signed up for a gift exchange (secret santa being one of many) you would get paired with someone else who signed up for the same exchange. You would then get a link to their profile where they enumerate their likes and dislikes and you would get an address to send your gift to. It will probably be the only thing I miss about Reddit. I'm seriously liking hubski from the little bit I've seen though.
We've done book exchanges and I think record exchanges. Since the community is much smaller, it was "someone picks out of a hat" style rather than a organized dedicated website. If you are interested, I would say start an informal one yourself and see if it generates interest. I would recommend waiting a couple of weeks until you see who sticks and leaves from this influx. Traditionally, some people only stick around for a couple of days. If they're here after a couple weeks, they're usually here for life.