If you want a detailed account of the philosophical issues raised by the Star Trek teleporter, you should check out CGP Grey's The Trouble with Transporters, a YouTube video that came out earlier this month and has been zipping around the Internet since then. It's smart, it's fun, it's short; you should watch it.
But my favorite film about these issues is a lot older than Grey's video. It's a cartoon John Weldon made in 1990, called To Be:
For yet another video on the subject—this time from a physics angle—go here. For a novel that touches on the topic (and later was made into a movie that I'm told is good but I haven't seen), go here. For past installments of the Friday A/V Club, go here.