Place of registration and place of actual residenceLike in most countries, government services are offered by place of registration. I live in Moscow, so I get my passport in Moscow, not in, say, Novosibirsk. Although…, in Russia the system is just a little bit different :)
Place of registration and place of actual residence can, and quite often do, differ. This is seen as completely normal. E.g., when I opened a bank account, I was asked for my place of registration and for my place of actual residence, which at that time (we were renting) were in fact different.
The reason for this is because registering by place of actual residence isn't for everyone. You need to either own the property you want to be registered at, or you need to get the owner to register you for you. Landlords aren't very keen on registering their tenants, because it gives you right of usage. In other words, they can't kick you out.
It thus so happens that a lot of Muscovites, who rent here, have their place of registration in some other part of this enormous country. In order to apply for, say, a new passport, they'd have to drive or even fly back home, and apply there. (You can apply for a passport elsewhere, e.g. by place of actual residence, but then it'll take four months instead of one. Yes, months.)
RegisteringThe actual process of registering, though, is quite easy. (Compared to the entire immigration disaster. As a Russian, everything is just so much easier :P)
I went to the so-called passport office with my mom-in-law (the owner), filled out a form, handed in flat documents and copies, as well as my passport, and that was that. In a week, I can come pick up my passport with a fancy registration stamp in it! :)
Btw., about it being easier as a Russian, it is. When I got my permanent residence permit, I also got registered, much the same way, except back then I needed written permission from all those already registered. Even if you co-own the flat, you still need this permission.