If you want to read the Wikipedia entry on APIs, knock yourself out. You might also find the entry on Web Services interesting. I'll wait here.
Done? Okay, welcome back. There's a lot of really dense information over there, and I find it useful to fasten on two key concepts: One, an API is a way for computers to talk to one another, and two, a web service is (kind of) a web site with the skin taken off.
Remix, for example (hey, what other example am I going to use?), replicates pretty much what you can get from BestBuy.com on the "product detail page" for a given item. Here's an Insignia TV at BestBuy.com:
And here's the same TV in the Remix API:
Notice that pretty much all of the information that is in the pretty BestBuy.com version is also in the texty Remix version? That's because Remix is more or less just BestBuy.com with the skin off.
So what you're seeing is information from BestBuy.com, but not at BestBuy.com. When prices change on BestBuy.com, they'll change in that sidebar gadget.
What do we hope to get out of this? Something cool that we can't imagine. Yellow Tag Productions, our internal video production team, did a great video that summarizes the thinking behind this and the potential we see, called "Open for Business." It's worth 4:50 of your day, I think. Of course, I would, since I'm in it. Give it a try.