Most any European phone, and those from the US which work on a GSM network (T-Mobile, AT&T or Sprint), carry the "tri-band" or "World phone" designation and had been unlocked, should work on the Russian standard.
Major cell phone providers in Moscow often sell SIM cards fairly cheap so it is possible to bring your phone from abroad. If you buy a SIM card from a shop you'll need your passport for identification. It only takes five minutes to do the paperwork and it will cost less than $10. You will receive a number in the "mobile" area code, starting with 9, which has more expensive rates for calls to and from landlines.
However, if you are planning on making a long term move to Moscow, buying a local pay-as-you-go SIM card instead of paying for roaming is more cost effective. Incoming calls are free (although some companies are trying to find ways around this). More minutes can be purchased at the stores of your chosen company, at shops selling phones, or at newer automated kiosks which accept utility payments (they appear similar to ATMs with large touch screens and they charge a small commission fee).
In Moscow there are three main GSM operators: