For single and double entry visas, you can register simply by having your sponsoring organization stamp your visa. You will need a Central OVIR registration as well, however, if you intend to marry in Russia or buy a car. If you are living only in a hotel during your stay, the hotel will register you. But count your stamps. Your visa will be stamped once when you check in, and a second time when you check out.
Some people think that the first stamp is sufficient.
It is not. What happens if you screw up You can be fined if you don't quite get your registration right and in some very rare cases can even be deported. When your visa expires Visas can be renewed for a week or two when they are expiring - long enough to allow you to finish what you're doing and leave the country. If your visa can not be renewed you need to apply for a new one. This usually necessitates a trip out of the country, although some agencies do offer visas that don't require you to leave.