A pingback is a notification that someone else has linked to your site. This form of linking back to your site is known as many different things, like trackback on WordPress, or backlink on Blogger, but they are essentially the same thing.
Pingbacks are there to link together posts so if blog A posts something interesting, and blog B posts a response to this, or develops a point in the original blog post, and links back to the blog post on blog A, then when someone views the post on blog A, they’ll see a link to the post on blog B, which will probably be of interest to them.
If you’re writing a blog post that was inspired by another persons post, then it’s polite to link back to them. It’s also a good way to give the other blog a little SEO boost if you link to their post with specific wording although be careful not to sound too spammy otherwise other bloggers may choose to disavow your links with Google. (Disavowing is when you tell Google to disregard links from certain sites when calculating your PageRank.)
Pingbacks are really excellent to receive as a blogger though, not just because of the benefits to your blog, but because it can introduce you to blogs you might not have heard of before!