Operation Y is like a legendary russian comedy.My parents and grandparents lived in USSR and that movie was shown soo often and so many people loved it that many of them know this movie by heart(my family included).There are some other legendary russian comedies but I'm not sure about their english titles now.
For crime movies,most popular are Brother I and II (the main actor died in 2002
() and Bumer.Also if you like organized crime or mob stories,there is a very interesting tv show "Brigada" (Brigade).There are about 15 episodes,it's definitely worth seeing.
for dramas,I've seen Lilja 4ever and Malenkaya Vera and they looked very realistic to me.Family relationships,music,lifestyle,dialogues - very accurate,even nowadays such things happen.If you want to know more about the soviet lifestyle,values and young people,you should watch them.
As for the famous Tarkovsky - I liked Stalker but I didn't like the Mirror,it was too confusing.I'm planning to see Solaris one day (lithuanian actor is in the lead) and Ivan's Childhood looks interesting.
For war movies,many people love "9 rota"(not sure about english title).Well,it's quite sad but also a bit mainstream.Another popular war movie is "Svolochi" ("Bastards"?) but I absolutely disliked it,not sure why because I haven't seen it in ages.
I would also recommend "Idi I Smotri" (Come and See).Didn't like it a lot (I needed more character drama) but it's quite disturbing.
Also there is a great Russian movie about the Siege Of Leningrad,very disturbing and shocking but I can't find it,I don't remember the exact title. :/