WHAT IS THE REGISTRATION. After you arrive to Russia you need to register in 72 hours (3 working days - excluding public holidays and weekends) after your arrival. You yourself, the hostel or hotel you stay in, an inviting organization or a travel agency can make the registration. Doing it yourself is taking too much time and some money.
Registration is a small stamp in your passport on a paper attached to your visa saying that you're registred for living at some place for some period. Registration is needed because the city officials want to be sure you're living somewhere.
Until it's 72 hours after your arrival, your tickets (train, bus, plane tickets) to the place you're in (e.g. Moscow) is an official document which confirms for how long you've been staying there. That means the tickets can be used instead of registration for 72 hours upon your arrival. After 72 hours you need a registration.
If you stay in a hostel or a hotel they usually make this registration for you at the desk for free or a small fee ($1). But the hostels make reservations only for those visas they themselves made invitations for.
HOW TO REGISTER YOUR RUSSIAN VISA. If you made an invitation through a hostel, they'll register you for free, if you buy at least one night there.
Another way is to go on the first day of your arrival to any crappy cheap hotel (unfortunately, the even cheapest is for $18-20 US a night), apply for the room for one night only, and ask them to make a registration for the whole period of your visa. You can stay in this hotel one night and then do whatever you want and sleep wherever you want. If you want an address of some hotels that can do it for you, contact us.
Hostel "Asia" can make a registration for $7 US if you got your visa not through them.
If you got your visa from some travel agency and the hostel/hotel you stay in can't make a registration for you, you can go or call to the travel agency and they'll explain to you how to make this registration. Usually, if you live in an apartment, the owner of the apartment makes the paper where he says that he allows you to stay at his place from such until such date. Then he signs this paper and stamps it at the lawyer's office. After that you bring this paper and your passport to the travel agency and the agency makes you a registration for about $20 in one day.
WHAT IF YOU DON'T REGISTER / POLICE: This registration thing was made more for people from CIS rather than the foreigners. Some cops will care if you have a registration or not, some not. If a cop stops you to check your documents and you don't have a registration, he has the right (russian law) to take you to police station and to fine you.
In the worst case, if you don't have a registration, you might have problems in case police stops you to check your documents. Officially, the fine should be $3-5, and they can also take you to a police station, but for not long (not more than 3 hours).
But there are some fuckers who try to make bucks on foreigners and start saying that "Oh, you don't have registration, you're going to have big problems, you have to pay $100 not to get involved". All this is crap. These policemen are just trying to make money on you. Officially they can't arrest you or take $100 from you for such a minor thing. But they start to frighten and all to make money, as if they forget that they may lose their job for bribing. Usually, these fuckers like to work next to the busy tourist area (like Red Square in Moscow, for example).
What you can do is to either not pay anything (if you have time and patience to argue with them) or just put the price down to the standard $3-4 for such infringement. Better don't speak Russian at all and behave as if you don't understand what they want. One policeman told me that if they see a foreigner doesn't have a registration, they'll not care because they "don't want extra troubles".
CUSTOMS. If you get lucky and nobody stopped you, then you might have problems on the border. The customs will ask you for the registration papers and if you don't have any they can ask for a fine up to 100$ - guys in the travel agencies say it happens in 30% of cases. You can just say you don't have any money and hope they'll let you cross the border (it worked with a friend of mine), but in that case they could write down your passport details and you may have problems getting russian visa next time (they didn't write down the details of my friend, she came to Russia again after that without any problems).
And one more thing: actually most of the people don't register, cause it's pain in the ass, but if you want to feel safe and if you don't want to talk to russian police, it's better to register, especially that it's free or 5 bucks maximum.