Huh? Yes, it does. When you finish a song (that is a ranked/approved map), a little pop-up screen thingie comes up specifying about your ranking and such. Somewhere at the top there are options to save your play as local score and a .osr. You can also press F2 in order to export as a .osr. .Osr's are stored in the Replays folder in the osu! directory along with all of your songs, skins, screenshots, etc.
You can import .osr's by opening them with osu! or simply dragging and dropping into the osu! window.
However, osu! replays are not stored into the osu!DB if you somehow fail to rank in the top 40s in a particular beatmap. To view those replays, it must be stored onto the local scoreboard.
EDIT:
mattyu007 wrote:
He meant uploading osr's to youtube
Right... I guess I misunderstood there.