I assume that you are interested in the video and arcade games created by the Soviet Union and not smuggled/pirated console and PC games from the West. It's somewhat difficult to tell what was popular, as this article on soviet video games mentions, high score boards were not prominently featured in soviet video games, moreover soviet video games were significantly more expensive to play than western games leaden to less of an arcade culture than the west. However a list of number of Soviet games and there descriptions can be found here The Museum of Soviet Arcade Machines also hosts an emulation of a Soviet arcade game here

Not Soviet but Warsaw Pact nonetheless: a discussion thread on this sub last week lead me to a post about the Poly Play, an East German government-owned gaming cabinet by VEB Polytechnik that had multiple games; I had to know more! The thread was here, this is a post about Eastern Bloc computers. The removed comments were the anecdotes of someone who had grown up in the DDR and played a Poly Play at the Young Pioneer Palace in his home city. You can see their Pac-Man rip-off in action on YouTube here. Poly Play was the only arcade unit developed by the DDR.

There was a BBC article I found here, which discusses the Poly Play within the context of a functioning unit making its way to Britain. If you'd like to try your hands at the games, they are emulated online here; it's a laugh. Thanks /r/AskHistorians for giving me a little hole to climb down on a rainy night!

This kind of games were huge starting from 1984. Also there were clones of Western computers like ZX Spectrum and there were video games there. Almost no consoles though (cartridges were hard to copy, I guess).