Russian is closely related to Ukrainian and Byelorussian, these three making up the East Slavic branch of the Slavic languages. The other two branches, rather logically, are west and south. The West branch includes Polish, Czech, and Slovak, and the South branch includes Bulgarian and Serbo-Croatian, as well as almost all the other languages of the former Yugoslavia (pretty much everything that's not Hungarian, Albanian, or Greek). Slavic and Baltic, which consists of Lithuanian and Latvian, make up the Balto-Slavic branch of Indo-European or Indo-Hittite.
Indo-Hittite (which includes Indo-European), in turn, may be a branch of Eurasiatic, which includes several languages of Europe and Asia.