You must report to the local passport control (OVIR) within three days (holidays and weekends excluded) of your arrival if you where invited by a citizen of Russia. I believe if you come to Russia with a tour or booked in a hotel they will look after this for you, but check with your travel agent prior to your departure. Now when someone invites you to come, you must report to this office as they will stamp your visa that you have arrived. It is the old Soviet system that kept track of all visitors in Russia. Technically you can only be in Russia where your visa lists you can.

You can be arrested or detained if you are in a city not listed in your Visa. Most likely you will just have to pay a bribe to the authorities who stop you. But you can be deported. The problem with OVIR is they are only open at certain times, no English is spoken and no English signs to help you as to what you must do. In Moscow they may speak English but elsewhere forget it. It is up to the person who invited you to also see that you report in, so it is best you have them take you to do this. You will be required to drop off your visa to them, pay a fee (less than $1) at a Government payment location, then report back usually in three days or so. Give them the receipt that you paid and get your Visa back. Sometimes they are not so friendly, but over the last few times they seem to have become more easy to get along with and more helpful than in the past.