In most states it determines which primary you can vote in.
And how does anyone authenticate what your affiliation is if you appear at the polls during primaries? If one likes the president and has every intention of voting for him again would it not be fun to go and vote for Dean for example. I guess I still don't get it's significance?
If I were registering right now, and the lady said "Democrat," that's how I would register until the next election...but that's just me.