How do I find out how many backlinks my site has?

Pop onto alltheweb.com and type in a search for link:www.yoursite.com

It will come up with all the pages linking back at ya...

Google is a bit confusing when it tells you how many backlinks you have, try using this tool it will give you a slightly better idea: http://www.linkpopularitycheck.com

Yahoo seems to be pretty good at reporting lots/all links:

linkdomain:yourdomain.com