There is only one file type that uses the .OSR extension:
osu! Replay File
Game replay created by osu!, a rhythm game where players synchronize mouse clicks with music; saves a recording of a song played by the user, as well as the player name, date played, score, hits and misses, and ranking; often used for sharing replays with friends.
If you receive an OSR file, you can double-click it to see the replay summary and watch the replay (assuming the osu! software is installed).
Replays can be saved at the end of a game by clicking the save replay or export replay option. OSR files are saved to the \Replays\ directory of the osu! software installation. You can copy these replays and send them to friends over email or another file transfer program.
Updated: May 27, 2012