Good question. I'll try to give a relatively short response.
When weblogs first started, they were very simple, usually short posts, and the tools for creating these entries were really simple. It turns out that people like simple things, so the concept of a weblog grew in popularity. As more and more people began posting entires on their weblogs (by this time, they were just called blogs), people realized that short excerpts of personal communication beg for a response. At first, comments allowed for people to respond, and this worked fine for simple, short responses. However, people often really get into a dialog and want to make extended points. A natural place to post a longer response is on your own blog. So, let's say I respond to one of your blog posts by creating a post of my own, but you don't read my blog regularly, so you miss the fact that I just tried to engage you in an exciting "conversation." When this happens, what we need is a way for my blog to tell your blog that I responded to one of your blog entries. And, you guessed it! That's what a Trackback link is for. It's a link that someone else can use to send a ping back to your blog to let your blog know that someone has posted a blog post of their own in response to your blog post.
In short, if you approach blogging like having an online conversation, then Trackbacks provide a way for people to let you know when they've either responded to you or just referenced your post from their own blogs.
If you're interested in more information regarding Trackbacks/Pings, this might help.
Some blog clients (and the DNN blog is supposed to do this, but I haven't gotten it to work right yet) will process the contents of each new blog post when you save it looking for Trackback URLs which may have been included in your post. When it works, this is a nice feature. I'll be adding this feature to a future version of metaPost, which is a free module you can use to post entires to the DNN blog (and soon to other third party modules) from within Windows Live Writer.
HTH,
Don