How do I check how many backlinks are on my website?

[Jake]

Actually the search engines report a portion of the linking sites, Google is renowned for keeping their link data close to the vest ,though registering your site wit Webmaster Tools gives you access to more omplete data. Search for link:yoursiteurl

There are a number of services, typically giving limited free results

OpenSiteExplorer.org and BacklinkWatch.com are 2 well known examples.

They don't check every one of the billions of pages for links, more likely the junk pages not worth it anyway are skipped.

One more exotic tool.
http://www.majesticseo.com/reports/compa...nk-history