Pingbacks and trackbacks are similar enough for the words to often be used interchangeably. The main difference between the two is that they use different methods for communicating with the server that receives the notification. Pingbacks use the XML-RPC communication protocol, while trackbacks use the HTTP POST protocol. There are also occasional differences in how themes present the two.
Pingbacks and trackbacks are usually displayed in or nearby the comments field, although it has become increasingly rare for themes to display them at all. The decreased popularity of pingbacks and trackbacks has a lot to do with the issue of comment spam, which often come in the form of trackbacks. A spam filter plugin like Akismet can help alleviate that problem. All themes on the WordPress.org Theme Directory are required to support output of pingbacks and trackbacks.
You can manage the site settings for pingbacks and trackbacks in Settings → Discussion. Here, you can select whether to send pingbacks and trackbacks to other sites, and whether you want pingbacks and trackbacks sent to posts on your site to be displayed in the comments section. Both of the two options are activated by default in WordPress.
You can also deactivate pingbacks and trackbacks for individual posts if you open the post in Posts → All Posts, scroll down to the meta box labeled “Discussion”, and check or uncheck the checkbox labeled “Allow trackbacks and pingbacks on this page”. If you can’t find the box, scroll up to the top of the page, click the Screen Options tab and make sure that “Discussion” is checked beneath the “Boxes” heading.