I had mine done the night I turned 20, and I don't regret it yet. I can't really imagine regretting it either; any more than I could an eye or a hand. It's just part of me now.
Personally, I think that holding off doing something you really want to do because you might regret it in the future and for no other reason is a little silly. We do only live once, after all, and I think you should go for what makes you happy.
That said, I also think that the more permanent decision, the more you should consider it before hand. If you want to cut your hair on a whim, go ahead! It grows back, after all. But a tattoo is hard and painful to ever get rid of, so it's important to be sure of it first. If you've weighed the pros and cons, and you're as sure as you can be that it is something you want and that you'll continue to want it in the future, then go ahead!
So, I'd say the earliest age you should get a tattoo is whatever age you're mature enough to think rationally about things like that, and make such decisions responsibly. For most people, I think that age will be somewhere between 16 and 20; but everyone is different. For some it might be earlier, for others it might be never.