To come to Russia you need to get the Russian visa. A visa to the Russian Federation is a document, permitting stay in Russia for a specific period of time. A visa lists entry/exit dates, your vital information, passport details. You can obtain it in the nearest to you Russian Consulate.
In order to apply for a visa you need a set of documents (you can find the detailed list in the Russian Consulate). The necessary documents are: a valid passport, an application form with a photo and an invitation to Russia from the Russian company or a citizen.
Please note, that your Russian visa is an entry/exit permit, if you lose it or stay over your designated departure date, leaving the country could be more troublesome than entering it.
There are following types of russian visa:
1. tourist visa
2. business visa
3. transit visa
4. private or homestay visa
5. student visa
To apply for russian visa you need visa support documents.
On arrival you need to feel in the migration card at the airport and should be registered.
Here is a very usefull link on the website, where you can find more detailed info.
It is very important to know about some changes since 2002:
In November 2002, the Russian visa system underwent a bit of reorganization. Issuance of visa support was previously the responsibility of the Ministry of Foreign Affairs, with whom each Russian host organization was registered. This responsibility was transfered to the Ministry of Internal Affairs, specifically the UVIR/OVIR offices. These offices had previously been responsible for the registration of foreigners on the territory of the Russian Federation (obligatory within 72 hours after arrival) and so this is actually an additional function given to an already overburdened office.
If you are visiting Russia for business especially, be aware that not all Russian organizations are authorized to issue visa support. They may often use 3rd party suppliers and in fact much of the new legislation coming into force is intended to discourage this practice, especially as concerns long-term multi-entry visas.
If your visit is short, it is easiest to obtain tourist visa support and register by staying in a hotel. If you will be a frequent visitor to Russia, you should endeavor to stay abreast of any changes concerning both visas and work permits.
Note that upon arrival to Russia, you are given (usually on board your flight) two documents. One (a fairly new requirement) is a migration card. This will be stamped at passport control and half of it torn off. Keep the other half as you must submit it upon departure. The other document is for declaring any valuables at customs. If you are planning to bring any sizeable sums of money back OUT of Russia, it is best to declare it. If you have money wired to you at a bank in Russia and due to this need to bring out more than you bring in, you need to request a spravka from the bank at that time as that document shows the source of the money.
You will notice especially on multi-entry visas now that there is not only the visa pasted to your passport page, but also a white paper stapled to this. Finally it has become clear what this is for. When arriving at large international airports, passport control is computerized and so they do not need to enter all of your data. But if you travel across the border by train, they will tear this off (or half of it actually as it is usually 2 copies on one sheet). If you will be doing that often, make some extra copies of that white sheet. Also, note that when applying for a visa, you must have 2 opposing pages empty in your passport or they cannot issue the visa.