A razor blade works best for cutting. Clippers tend to crush the tissue a bit but they work fine too.
Just use a rubber band to secure it. Not too tight as it may cut through but tight enough so that it stays on the disk or rubble that you are trying to mount it to.
Edit: youtube.com has roughly 1000 videos that show up if you do a search on coral fragging and coral propagation. There is also this frag of the month article in reefkeeping about leathers.
http://reefkeeping.com/issues/2006-09/fotm/index.php