What do musicians use to play backing tracks?
DAW of your choice: Ableton Live is a fantastic option for playing backing tracks (among many other things). Other software options include LiveTraker and Digital Performer to set up your tracks for the show. Reaper is also great for multi-track playback.
Can I use someone else's backing track?
If you do have permission from the person who holds the copyright, then you can do it. Not the person who made the backing track, but the person/people who holds the actual rights to the song.