Trackbacks are a way of telling another blogger that you have found their post useful, and have made a post of your own which continues the discussion. So, rather than include a comment in the other's blog, you can include an excerpt of your own post with a link to the whole thing. This way people can read your own perspective on the subject.
So, how do I use trackbacks?
Well, in order for a trackback to work, both blogs must have the trackback feature installed. You can usually tell when another blog has the trackback feature enabled, as there will be a link beneath their post (usually near the "comments" section) which says "Blog this" or "Trackback URL".
To send a trackback, you will need to use the post's designated trackback URL, otherwise your trackback will not be sent and won't feature in the trackbacks section.
When you want to send a trackback notification to someone's post, you'll usually do this by inserting their trackback URL somewhere in your posting page which will send a ping to the other's blog when your post is published. In Wordpress, for example, this is just below the composition box. It's usually considered good manners to include a link to the post being trackbacked in the body of your own post, though not strictly necessary. Some bloggers also suggest a follow-up comment too.
But I have a Blogger/Blog*Spot blog! How can I use trackbacks?
Unfortunately, Blogger doesn't make use of the trackback feature by default. This decision may be due spammers abusing the trackback facility. However, there is a way Bloggers can still use trackbacks: by installing Haloscan.
Haloscan uses it's own system to manage and create trackbacks from its site. By creating an account, you can install Haloscan trackback and comment links into your Blogger template (either by hand or using their new widget feature) so others can send trackbacks to your posts. If you would like to send a trackback, you can do so from your Haloscan dashboard.
You can see how Haloscan trackbacks work by clicking on the Trackbacks link at the bottom of my post (I've kept Blogger's system for my comments). Though this may be a slightly time consuming way of using trackbacks, I still prefer this to not being able to trackback at all! I've written a tutorial to help you install trackbacks in Blogger templates here.
What about spam?
Unfortunately, spammers understand that it's possible to use trackbacks to create links to spam sites. Popular blogs may well be plagued by these nuiscence links, as spammers love to increase their popularity by having their links on sites with good page rankings.
You should be able to delete spam trackbacks from your blog manually, and may even be able to moderate all trackbacks before they appear on your blog. The only way to prevent spammers for good is to disable trackbacks altogether.
Using trackbacks can be a great way of getting others to notice your blog and form discussions across different blogs. But it should not be abused! Many bloggers are offended by trackbacks which seem to bear no relevance to their actual post, and delete them regardless, so as I mentioned earlier, it may be best to include a link or reference to the blog you are trackbacking in the body of your post. After all, the other author has taken the time and effort to write the post which inspired you!