How it works:
A pingback is sort of like a trackback, except automatic. (Okay, 80% of the audience just had their eyes glaze over.)
Pingbacks are based on the concept that if you link to someone else's entry in public (or if they link to yours), you'd want to let them know, and you'd want to know if someone else is talking about your entry. Even better, you want to leave a link to your post on their post to maybe get some of their readers over on your entry.
So you make a public post and you have pingbacks turned on (for your whole journal, or just that post). It has a link to somebody else's post. "Oho! A link!" LJ says, upon you posting it. "I wonder what's at the other end of that shiny little link?"
LJ then pokes the link with a stick to see if the other end of the link does the pingback thing. (If it's on LJ, it's easy to tell.
If the other site doesn't do pingbacks, LJ shrugs its shoulders and soldiers on. If it does, the other site goes, "Hey, thanks! ...uh, just to be sure, you really do have a link to me, right?" and prods your entry to make sure that you've got a legit link to them and aren't likely to be a spammer just saying you have a pingback.
Assuming all is good, the other site lets the blog owner know that hey! They have someone linking to them! OMG! and in short order, a little blurb saying something like "Some random guy at a blog you've never heard of linked to you! '...and in other news, I found a cool link: [link goes here] and zomg, what, and also ...'" shows up.
In turn, when someone else links to you (and you both have pingbacks turned on), a screened comment from pingback_bot shows up on your entry.
I recommend you turn this on if: you're participating in a large, wide-ranging discussion, and/or you wouldn't mind if the original poster and/or all their commenters followed the link back to chat with you in your post.
I recommend you turn this off if: you're making a catty public post, or any other situation where you would not want the original poster and/or all their commenters to join in or necessarily even know that the post exists.
It doesn't matter if: your post is not public -- pingbacks are off for all locked posts.
This post is brought to you by another round of multifannish discussion and friendly commenters coming in and mentioning where they're being linked from, and a general lack of pingbacks from the places that are linking. (Except you, 'song.)