Pingback allows you to notify other bloggers that you have linked to their article on your website. Although there are some minor technical differences, a trackback is basically the same things as a pingback.
If the other blogger or author has a pingback enabled website, then they can see a notification that you have linked to their article. They can then choose to allow your link to appear on their website. Depending on how your WordPress theme is configured, pingbacks on your site may or may not show an excerpt of content from the other person’s blog.
You can choose to allow or disallow pingbacks in the Discussion Settings of your Administration Panels. By checking “attempt to notify any blogs linked to from the article” WordPress will notify other blog owners when you link to them. In other words it will send them a pingback. If you check the box labeled “allow link notifications from other blogs (pingbacks and trackbacks.)” then you will receive trackbacks when other authors linking to your content. These will always show up in your admin panel where you would normally find comments. Whether they display on individual posts is determined by your theme and not your WordPress settings.