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First of all, Cyrillic (what you call "the Russian alphabet" is not originally Russian, it is rather Bulgarian.
At the modern time, Cyrillic is used by the following languages:
1) some of Southern Slavic languages (Bulgarian, Serbian, Macedonian) - modern Serbian uses both Cyrillic and Latin interchangeably;
2) all Eastern Slavic languages: Russian, Belorussian, Ukrainian;
3) During the Soviet era, all the national republics of the former USSR (which became independent states after 1991) used Cyrillic for their national languages, with the exception of Latvian, Lithuanian, Estonian (they have always been using Latin), and with the exception of Georgian and Armenian (either of those two languages has its own traditional script).
After the dissolution of the USSR, some of the newly independent nations switched to using Latin: Moldavian (often believed to be a dialect of Romanian), Azerbaijani, Turkmen, Uzbek and Karakalpak (an ethnical minority language in Uzbekistan).
However, some of the former USSR nations are still using Cyrillic: Kazakh, Kyrgyz and Tajik.
Ukrainian and Belorussian, being Eastern Slavic languages, use Cyrillic traditionally (see 2).
4) all the ethnical minority languages of the Russian Federation:
Iranian languages (Ossetian);
Turkic languages (Tatar, Bashkir, Chuvash, Yakut, Tuvan, Altay, Khakas, Balkar, Karachay, Kumyk, Nogai and some others);
Mongolic languages (Buryat, Kalmyk);
Tungusic languages (Even, Evenk, Nanai, Udege);
Uralic languages (Mordvin, Mari, Udmurt, Komi, Karelian, Khanty, Mansi and some others);
Samoyedic languages (Nenets, Selkup);
Northwest Caucasian languages (Abaza, Adyghe, Kabardian) including Abkhaz which is spoken outside of the Russian Federation;
Northeast Caucasian languages (Chechen, Ingush, Avar, Dargwa, Lak, Lezgian, Tabassaran and other languages of Dagestan);
Chukotko-Kamchatkan languages (Chukchi, Koryak, Itelmen);
Eskimo-Aleut languages (Aleut, Yupik);
Separate isolated languages of Siberia (Ket, Yukagur, Nivkh)
- the list maybe incomplete.
5) Cyrillic has been used outside of the USSR area by Mongolian language since 1921, and is currently in use in Mongolia.
