API = application programming interface, a standardized way for programs to make requests and receive information. Today if you wanted to talk to Google programmatically, your program would have to spoof being a browser, submitting HTML forms and parsing the HTML results, which is both tedious and annoying to implement and very fragile 'cause it'll break the first time Google changes anything about their pages.
People would program with Google so that they can take advantage of Google's indexing of the web to do any of a bazillion different kinds of data-mining... the possibilities are endless. What are the commonest words on the web, and how has that changed over time? Track the rise and fall of concepts or celebrities through their frequency of reference over time. Map the net through social networks a la blogdex (or in any of a zillion different ways.) See what the shortest distance is between two sites. Build specialized search engines to facilitate searching on given topics or in given ways. Perform any of a zillion kind of censuses of the web: how many Baptist sites are there? And these are just some of the obvious things off the top of my head -- there are bound to be a bajillion less obvious really exciting things the hackers will come up with.
SOAP =
Simple Object Access Protocol, a W3 Consortium standard for abstracting data objects in XML intended for building APIs that'll facilitate communication between programs running in unlike environments.
As for money, Erik seems to be speculating they'll charge for access to their system via the API -- not a bad bet given how intensively an application might hammer it.
