The API is something that allows developers to write third-party tools to interact with the company's code or database. Basically, an API lets other people write software besides just you.
Companies that have open API's tend to see an avalanche of third-party tools develop. For example, you know how almost every web site you go to uses Google Maps as the engine that drives their maps (store locator, jogging trails, etc.)? That's because Google Maps has an open API that is easy to work with. There is a bit of irony in noticing that Groundspeak switched to Google to power their cache maps several years ago. Why were they able to do this? Because Google Maps has an open API.
Groundspeak on the other hand, has a history of actively disallowing people access to their data for making third-party tools. One of the very, very, very, very first mapping applications for geocaches was developed by a guy named Ed Hall. He created Buxley's maps, using cache data from geocaching.com. There are a few exceptions that they've carved out, mostly involving giving permission to a few applications to accept PQ uploads (GSAK, mygeocachingprofile.com). But those are very much on a case-by-case basis and are pretty restricted. Those sites aren't allowed access to groundspeak's data themselves, for instance; they can only take in the limited PQs that a user can generate. Interestingly enough, Buxley's maps still exists as a site, although it claims to carry no groundspeak listings -
http://brillig.com/geocaching/new_york.shtml .
Know why there aren't any legitimate competing iPhone apps for geocaching? Because groundspeak's API is closed and they won't allow anyone else access to their data. It's a good strategy for keeping a large percentage of the pie for yourself, but not so good a strategy for growing the pie. They've moved pretty aggressively to block people who have come up with workarounds (like interfacing with the servers through the web).
As one contrasting, simple example... Opencaching.com uses Bing maps. I don't really like Bing maps. Within the first 48 hours that Opencaching.com debuted (with their totally open API), one of the guys in the groundspeak forums whipped up a really neat applet that lets you pan around for OC caches using Google Maps and various filters for cache types and ratings. In hours he created something that was arguably better than the map function that groundspeak had a several years head start on! How? Because OC and Google both feature open APIs. It was a cool demonstration, but it was quickly deleted by the moderators.
I'm under the impression that groundspeak was at least considering opening their API up a bit before, as it was the highest vote-getter on the "Feedback" site. But the timing of their announcement made a lot of folks feel it was in reaction to OC.com.