Registration of visas is a form of migration control within Russia managed by UFMS (Federal Migration Service Organisation). It is not aimed specifically at tourists or business people, rather its general aim is to control illegal immigration from the former Soviet Republics into the Russian Federation, though as the law is all-encompassing, foreigners including tourists are also required to register.
Basically, registration involves placing a stamp either in your passport or on your migration card (link to) and obtaining a registration slip. The stamps and registration slip show the period you are registered to stay in any one place. In addition to the dates it will also contain the name of the hotel, accommodation or apartment, where you are staying and the name of the sponsor organisation.
Please be aware that the Russian militia often stop people to check their passports and travel documents, especially around train stations, metro stations, bus stations and areas such as the Kremlin, so remember to take your passport, migration card and registration slip with you every time you leave your hotel or apartment. There is nothing unusual in this and there is nothing to be worried about if you are stopped, so long as you have registered correctly.
If you are stopped by the Militia for any reason, they will check that you are registered to be in the city / town and that the dates are valid.
New rules implemented from 25th March 2011 mean that you should register within 7 working days of arrival, rather than the previous deadline of 3 working days.
It is the responsibility of the visitor to register in each place where you spend seven working days or more. When registering in each new city (where your stay is seven working days or more), you have to hand in the registration slip from the previous city, so we recommend taking a copy of all your registration slips. You must hand in the final original registration slip at the border on departure from Russia. In St Petersburg, however, registration is usually completed by hotels and agencies through an online system so you may be advised by the hotel/agency to keep the original registration slip from your previous city of stay (where applicable) rather than handing it in to the St Petersburg authorities; in that case, you should hand in both original registration slips at the border on exiting Russia.