You're not alone, I'll tell you that. I was recently attending a blogger meetup here in Colorado and was rather surprised to find out that many bloggers don't understand the difference either. When one attendee endeavored to explain it, I realized that he too didn't fully understand.
Here's the difference: a ping is a message sent by a blogging application to blog search engines like Technorati that inform the systems that you have posted a new entry. If you think about how a regular search engine like Google works, you'll realize that when you add a new page, nothing happens until the search engine stumbles upon it, so the "on demand" pings are quite efficient and are one reason that blog entries are so quickly indexed. If your blogging tool doesn't send out pings -- which most do -- then you can also generate these system pings through a service like pingomatic.
By contrast, a trackback is a mechanism that attempts to have two blogs engaged in the same discussion threading their discussion so that someone who stumbles upon one weblog will see the conversation continued on the other. For example, let's say Mike Arrington at Techcrunch and I got into a debate.
On my weblog, I could then include a link to a specific blog entry on his site that I'm addressing. By doing so, my blog system would then automatically send a trackback to his weblog system, and if he'd configured it to accept trackbacks, his site would automatically publish a link back to my blog entry and discussion, as if it were a comment left on his blog. If he then blogs about what I've written and references my weblog entry, I'd see a trackback from him. Etc etc.
The theory of a trackback is splendid, but the reality, the pragmatic reality, of trackbacks is that unfortunately they were quickly overrun by spammers who realized that it would be quite easy to send spurious trackbacks and gain links to their junk sites from legitimate weblogs. As a result, most bloggers have trackbacks disabled (I do, for example) and we all just accept that there's not yet a good solution to the problem.
I hope that this clarifies the difference between a ping and a trackback.
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Dave, you are correct in explaining "ping" system that goes out and lets aggregators like technorati, odeo, and google blog search know, however there is a whole different "protocol" for communicating with other blogs, similar to trackbacks called pingback. It is built on top of XML-RPC protocol and is used to let other bloggers or sites know that you have linked to them. Here is the spec: http://hixie.ch/specs/pingback/pingback. Main proponent of it is Wordpress and I imagine there are plugins for other systems out there. Chances are if your blogging platform supports xmlrpc, it will support pingbacks. I believe the question asked at that meetup was the difference between trackbacks and pingbacks rather than trackbacks and pings. All 3 are equally important I believe. Although you have video evidence, and I'd imagine you can pull my punk card right back with this. :)