So, here's what these words mean, succintly:
This is the catch-all word. Pingbacks and trackbacks (and refbacks) are all linkbacks. Each of them offers a way to let the author of content published on the web (say, at a website or a blog) know when someone else has linked to their content.
Linked? As in "hyperlinked." Someone, somewhere has read what you've placed on the web, and they've taken the time to link to that content in their post, article, or other form of website content. For an example, I'm linking here to Wikipedia's definition of linkback, which I've found helpful in writing this paragraph, and which provides more details on linkbacking if you wish to go there and learn more.
If you are writing a blog, your blogging service provider (WordPress, Moveable Type, Blogger) may provide an automatic pingback service. There are also free services like PingOMatic and PingOat that do the same thing. Whether you rely on your blogging service or do it yourself, pinging insures that each time you write an article or post, that places like MyYahoo! and Weblogs and Technorati (i.e., blog directories) know you've just added something new. (Check them out, they don't necessarily ping the same places.)
(There's a lot of technical stuff about pingbacks, too -- things like they are sent via a distinct communications technology (XML-RPC), etc. but we're not going into that here.)
When someone links to your blog post, within your comments section there may be some strange type of comment that mimics your own lingo in your earlier content. That may well be a trackback, which is letting you know that someone, somewhere has linked to your work. It will give you a means to go read what they've written (usually the url to their blog post, etc.)
It gets confusing -- some will say you've received a "trackback ping" when this happens, and just stop yourself before you respond, isn't it a "pingback" then? There's a lot of techno detail here, "pinging" to some being any "ping" to your post.
Think of a magic wand, anytime it touches your blog, "ping" is heard. That wand may mean an individual blogger linked to your post and you've been notified in your comments section in a "trackback," or it may mean that a pinging service has notified all the blog directories within its service realm that you've posted something, in a "pingback."
Finally, while some services (like Blogger) automatically allow trackbacks, other services (like WordPress) may require you to decide if you want to allow them or not. It's tricky, and you'll need to check the details of the blogging service you've chosen.
Do you want trackbacks? Yes, you do. Why? Because they let you know when others are referencing what you've written in some way -- in their posts, in comments somewhere, etc. And not only is this something that lets you know you're being read (great!), it's also important to know how your work is being perceived. Trackbacks let you know what some folk are thinking about your contributions, and this is important, be it an accolade or constructive criticism. Trackbacks provide feedback, and that's good.
Image above: Google Blog Search, the blog directory for Google, Inc. Make sure your blog is included within the Google Blog Search by adding it manually. (This is a linkback to a pingback service.)