A ping is what happens when one software program notifies another one of an update.
Pingback - Pingbacks were designed to solve some of the problems that people saw with trackbacks. A Pingback is an XML-RPC request sent from Site A to Site B that requires a link. Some weblog software, such as Movable Type, WordPress and Community Server, support automatic pingbacks where all the links in a published article can be pinged when the article is published. Code on linking server examines added or updated documents, extracts links, and sends notification to linked server for each link found.
Pingbacks are transparent to the linking author, requiring no user intervention to work, and operates on principles of automatic discovery of everything that it needs to know. For instance wordpress is automatically configured to give pingbacks when you link to another post in your post the other person will get notified that you have written about their post.
LinkBacks - Allow you to notify another website that you wrote something related to what is written on a specific page. This greatly improves the chances of contributors to this page noticing that you gave them credit for something, or that you improved upon something they wrote. A linkback is a method for Web authors to obtain notifications when other authors link to one of their documents. This enables authors to keep track of who is linking to, or referring to their articles. The three methods (Refback, Trackback, and Pingback) differ in how they accomplish this task.
Refback - A Visitor to the linking site clicks on the link, and their browser takes them to the linked site. A Refback is simply the usage of the http referrer header to discover incoming links. Whenever a browser traverses an incoming link from Site A (originator) to Site B (receptor) the browser will send a Referrer header indicating the URL from where the user came. Site B might publish a link to Site A after visiting Site A and extracting relevant information from Site A such as the title, meta information, the link text, and so on.
This is an easier method than Pingbacks since the Site originating the link doesn't have to be Pingback enabled (Posting a link back within any webpage is good enough).
Trackback - Code on linking server examines added or updated documents, extracts links, and sends notification to linked server for each link found. The TrackBack specification was created by Six Apart, which first implemented it in its Movable Type blogging software. The TrackBack has since been implemented in most other blogging tools.