I've done a Google search on how to do this and found nothing. Surely there must be some search engine sort of tool that can tell me what websites have a link to my domain? Thanks
You mean backlinks? Yes, there are lots of website that offer such thing for free. Just Google it and type Backlink checker. If you have just made new backlinks, it might be turned live yet and you should wait quite some time for it to appear.
I'll put in a word for Ahrefs.com I'm subscribed to Majestic's basic plan which only shows historical, whereas Ahrefs supplies fresh index data on their basic plus a few $$'s cheaper.
We believe Alexa offers the top 100 sites linking back to you. Just type in your website URL and look for the Sites Linking In area. Just a thought if you don't want to use any of the other options above.
You can use backlinkchecker , iwebtool, seo spy glass, webseo to check all your inward link base on country and type of search engine.
you can use yahoo site explorer, type in competitors url and click search, then click on links and if you have seo quake installed load it and it will give you the top 1000 backlinks all you need to do now is arrange them in PR order - using quake tooolbar
Google webmaster tools - if you sign up for a new account you'll have to wait awhile before the backlinks start showing up, bing webmaster tools, yahoo site explorer, blecko, SEO spyglass - if you want more data than just the URLs of sites linking to yours.