Get in the top 50 of a song, and anyone can watch your replay.
Or just upload your file to puush/mediafire/dropbox etc and give the link to the person you want to watch it.
You can save your replay to the special replay folder by pressing f2 on the results screen or right clicking on your score (this almost never works so use the f2 method) and selecting export replay.
C:\Program Files (x86)\osu!\Replays
I.. think that's about everything =/