Pingbacks are those little snippets of text that get inserted into your comments section whenever someone links to your blog from another site. You can turn them on or off and, I believe, they need to be enabled on the originating site as well. For some folks they can be really useful as a way to both see and show to others who's linking to their blog. It's similar to the concept of followers on Twitter. It's a record of who's talking about you.
The problem for me is I almost never get pingbacks, with two exceptions. The first is spam. On occasion I've gotten pingbacks that were spam. This has mostly been cured by installing some anti-spam plugins. The other exception, however, is me. I link to myself all the time. And every time I do I get a pingback in the linked post. I don't really want these there, though, so I have to go and delete them. It's a pain.
You would think Wordpress would have a setting to block pingbacks from itself. Alas, I can find no such setting. Since I almost never get pingbacks from others, I think I'll go ahead and turn them off. The annoyance to usefulness ratio is just way to big.