Pingbacks
Pingbacks were designed to solve some of the problems that people saw with trackbacks. The official pingback documentation (http://www.hixie.ch/specs/pingback/pingback) makes pingbacks sound an awful lot like trackbacks:

For example, Yvonne writes an interesting article on her Web log. Kathleen reads Yvonne's article and comments about it, linking back to Yvonne's original post. Using pingback, Kathleen's software can automatically notify Yvonne that her post has been linked to, and Yvonne's software can then include this information on her site.
There are three significant differences between pingbacks and trackbacks, though.

Pingbacks and trackbacks use drastically different communication technologies (XML-RPC and HTTP POST, respectively).
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.
Pingbacks do not send any content.
The best way to think about pingbacks is as remote comments:

Person A posts something on his blog.
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.
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 pingback is generally displayed on Person A's blog as simply a link to Person B's post. In this way, all editorial control over posts rests exclusively with the individual authors (unlike the trackback excerpt, which can be edited by the trackback recipient). The automatic verification process introduces a level of authenticity, making it harder to fake a pingback.

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.