Belorusskie Shakhmaty ("Byelorussian Chess" - Белорусские шахматы) is a cross-over between chess and checkers. It is actually more a chess variant because its goal is to checkmate the opponent's king, but it also has rules borrowed from Russian Checkers. The game was invented by N. N. Grushevsky and P. A. Shkludov in 1984. The second author has published some problems for the game in newspapers in Minsk in the mid-1980s. The name on chessvariants.com ("Byelorussian Cheskers") is a mistranslation, but the rules are correct.
The game has rules similar to Russian Checkers and Chess.
That is, the checkers move and capture like the men in Russian Checkers and promote like them. The damki (дамки) move and capture as in Russian Checkers, too.
The chess pieces move and capture like the pieces in chess.
The pawns promote when they advance to the eighth rank and become a queen, rook, bishop or a knight of the same color. After that they move and capture as the respective chess piece (for the complete chess rules see the link below).
The player that checkmates the king of the opponent wins the game. Stalemate, check, etc., are as in usual chess.