A tourist Russian visa is a document that allows entry to Russia for up to 30 days. Usually it's single-entry only, but if one travels to one of CIS country (former USSR countries, excluding Latvia, Lithuania, and Estonia), it can also be double-entry.
In order to get a tourist Russian visa, a visa support (invitation) is necessary. It can be obtained from a hotel where you're planning to stay or from a travel agency. In case you get it through a travel agency, your visa support will have several hotel reservations in it, that's why it usually costs about $30 US. You are not obliged to stay in these hotels though, that's why a lot of people get their visa support this way. It usually takes about 24 hours to get visa support documents, and usually a faxed or printed out copy (from an e-mail) is enough, although some consulates require originals (Sweden, Switzerland).
Once you get your tourist visa support , you can apply for a Russian tourist visa by yourself at the Russian consulate in your country, or use one of your local travel agencies to do the application job for you (they'll usually charge about $20-$30 on top of the standard consulate fee for that). The current consulate fee is about 35-50 eur for EU citizens and about $100-$150 US for US citizens. The processing time is about 10-14 working days. It's also possible to do it faster, but the consulate fee will be higher then. Some countries receive visa applications by post, check with the consulate's website for the latest regulations on that.
Normally, in addition to the visa support, the consulate also needs your passport (original), 2 passport photos, and a filled out application form (available at the consulate or online). Consulates in some countries also require travel insurance from one of the approved provider (normally a major agency or your credit card insurance will do). UK citizens don't need the insurance as Russia has reciprocation agreement.
Also note that you can apply for a tourist visa only in the country of your residence (or where you have a permit of stay for longer than 90 days). So, EU citizens can apply in any EU country, but for example they can't apply in Beijing unless they have a permit of residence there. An application for the visa support can be done at any time, however the actual visa can only be obtained not earlier than 45 days before the actual travel.
Information provided by Way to Russia Guide to Visa