For most experienced bloggers, this is not news. However, for you beginners out there, this can be a very valuable skill/tool. Blog pros usually spend up to an hour a day digging through their website analytics. I know, that sounds absolutely insane, but it's true. Pro bloggers are blogging for a living, so it is very important that they understand every heartbeat of their site.
One (of many) ways to track the success of your site is to see how many other sites are linking to you. Google and other search engines actually look at these links as implicit endorsements of your content. Them more links to your site there are, the more "important" search engines believe your site to be.
Naturally, you'll want to have as many high quality sites link to you as possible. When they do, a quick "thank you" note back is always appreciated. Also, linking back is encouraged if you find thier site valuable as well, however it is not required.
Also, when a site links to yours, one way to digitally thank them is to head over to their site, browse, and comment where appropriate.
So how do you know how many sites are linking back to you?
Look at your site analytics. Hopefully, you have some sort of system set up to track your site usage. This allows you to see how many individuals are coming to your site, and how many pages they look at. It also allows you to see how they reached your site. If they clicked a link on a site to get to yours, it should show up in your analytics. If you don't have a stats package on your site, Google offers a good system, and it's free. Your web host probably offers pre-installed systems as well such as AWStats.
Technorati. After your site has been kicking around the web for a month or more, it will likely be crawled by Technorati.com, especially if it has been linked to by another blog. You can go there and do a search for your site. Technorati will then show you your "Authority" rank. That number represents the number of blogs that currently contain links back to your site. Technorati updates their numbers regularly so you can accurately track the links back.
Technorati also shows you a brief excerpt of each post where your site was linked. This allows you to quickly and easily browse the context in which your site is being talked about. You can also compare your rank to others blogging on the same topic.
Google Blog Search. Blogsearch also crawls blogs constantly, and will tell you how many sites link to your blog if you know how to ask. Simply log onto Google Blog Search (http://blogsearch.google.com) and do a search like this:
Type this into the search field
By typing "link:" in front of your site address in the google search engine, Google will return all of the sites that link to that address.
What if your site doesn't show up at all in Technorati or Google?
Perhaps it is too soon. If you've only been blogging for a few days or weeks, your site may not have been ingested into the index just yet. Or, perhaps no on has linked to your site yet. You can speed your inclusion up a little bit if you can get an established site(s) to link to your blog. That will draw the search spiders into your site.