When you move, you have to de-register at your old place of abode and register at the new place (though you can combine the two if you move within one municipality; then you just have to register at the new local office and they'll inform the local office responsible for your previous address). That will also update your identity card (they put a sticker with the new address on the appropriate area).
Since identity cards are compulsory and bear an address, they're occasionally used as proof that you live in a certain place (for example, people might take down your address when you rent a car or buy a mobile phone or something).
Since I'm not a German, I have no such address-bearing ID. When someone needs my address, I need to give them a "Meldebestätigung" - a confirmation that I'm registered at a particular address.
In the past, I'd typically get a new one whenever someone needed my address, since that didn't happen that frequently, but recently, I've taken to carrying a photocopy around with me along with my passport, so that I'll have both pieces together (photo ID + address).
Since we've moved, I also got a new confirmation of registration for the new address, and while putting a copy of the new one with my documents and removing the old one, I got a bit nostalgic: the old one is probably the last one where we have three different dates on it.
You see, the confirmation lists all of the people who live in a particular place, along with the date they moved in, and we moved in on three different days: I when the place was finished and ready to rent, Stella six weeks later after we had married, and Amy when she was born.
(Strictly, we all moved into our last flat together, but since it was one floor down from the previous one and the address didn't change, we didn't re-register. Nor do I think we could even have.)
But now, it says "18.10.2008" for all of us. Ah well.
Hm, and I just noticed that the new one only has one given name for all of us, while the old document had both of Amy's given names on it. Incorrectly, I suppose, since the title of that column is "Rufname"; that is, the name you're called by (typically the first given name, but in some cases, a later given name -- e.g. for my newest nephew, his third given name). And Amy isn't called by her full name, so just "Amy" is what it should say for "Rufname".
Speaking of which, apparently Stella's ID card is incorrect, since it doesn't list her maiden name. But it'll expire next May anyway, so she can get that fixed then.