When you arrive in Russia or when you change city in Russia you normally have to register yourself within 3 workdays. If you stay at a hotel, they automatically do the registration and it’s free. If staying at a hostel, they normally charge you if you want to get registered (e.g. at a hostel in Irkutsk it is 500 Rub).
When you stay with a private person, this person would have to go to the post office and fill out a form to get you registered.
There are many different opinions and reports on the internet about what would happen if you don’t register yourself. Many say you would have to pay a small fine if the police stops you, but in the worst case they could also deport you. Then some rumors state that it will be checked at the border, when you leave Russia, if you were registered, and you pay a very high fine if you were not, but other rumors state that you would not be checked at all.
Well here is my experience: I only registered myself in Irkutsk to have at least one paper to show. I probably should have done it in Moscow though. However at the border nobody asked anything about registrations, and I didn’t have to show my registration paper (they might have checked it on their computer though). I also know about other people crossing the border without registrations but no questions asked.