Codex, the WordPress help site, has a good explanation of how trackbacks work:
1. Person A writes something on their blog.
2. Person B wants to comment on Person A's blog, but wants her own readers to see what she had to say, and be able to comment on her own blog.
3. Person B posts on her own blog and sends a trackback to Person A's blog.
4. Person A's blog receives the trackback, and displays it as a comment to the original post. This comment contains a link to Person B's post.
Pingbacks are different and here is how they work:
1. Person A posts something on his blog.
2. Person B posts on her own blog, linking to Person A's post. This automatically sends a pingback to Person A when both have pingback enabled blogs.
3. Person A's blog receives the pingback, then automatically goes to Person B's post to confirm that the pingback did, in fact, originate there.
The Difference Between Trackbacks and Pingbacks
1. Pingbacks and trackbacks use drastically different communication technologies (XML-RPC and HTTP POST, respectively).
2. Pingbacks support auto-discovery where the software automatically finds out the links in a post, and automatically tries to pingback those URLs, while trackbacks must be done manually by entering the trackback URL that the trackback should be sent to.
3. Pingbacks do not send any content.
Some feel that trackbacks are superior because readers of Person A's blog can at least see some of what Person B has to say, and then decide if they want to read more (and therefore click over to Person B's blog). Others feel that pingbacks are superior because they create a verifiable connection between posts.
Setting Up Trackbacks and Pingbacks in WordPress
1. In the Dashboard, go to the Settings section and click Discussion.
2. The Discussion page appears. Make sure "Allow link notifications from other blogs (pingbacks and trackbacks)" is marked.
3. Click Save Changes at the bottom of the page.
Getting Notified of Trackbacks and Pingbacks
Once the settings are activated, you will receive an email when a pingback or trackback occurs if you have also set the Email me whenever settings appropriately.
The trackback or pingback will also appear in the Comments section of the post.
Should I Care About Trackbacks and Pingbacks?
Definitely. First, if a more authoritative blog (one with a large following such as Eastman's Online Genealogy Newsletter) links to one of your posts, it will help channel a large group of viewers to your site. And it improves your blogosphere reputation and "street cred." Second, you should always see who is linking to you and weed out splogs and other nefarious site to which you don't want any connection.