The actual API the Guardian has developed consist of two components: Open Platform and DataStore [guardian.co.uk]. The Open Platform will enable users to weave Guardian material, text, audio, video, and more, in online mash-ups, even commercial. In turn, Data Store offers access to about 80 data collections, which have been put together by the paper's editors. Their topics range from what people eat, over the smoking figures for every country, international suicide rates, or the number of plastic bags imported and exported worldwide.
Each Data Store dataset seems to be stored as a Google Spreadsheet, which means developers can access it using Google's Spreadsheets Data API. (Note: for those interested in more publicly available datasets, it might be worth it to also look at the Public Datasets on Amazon Web Services).
As a side note, some reports point to the fact of The Guardian shareholder structure making this possible: "It helps that the paper is owned and run by a charitable trust which does not have shareholders who would normally have a heart attack at such a move."
Some early application examples include Zemanta's Guardian topic researcher, which searches the database of Guardian articles and then proposes links to related concepts. There is also Content Tagger, an application to provide user-generated tagging on The Guardian content; ApiMaps.org, built by Stamen Design, designed to crowdsource geodata based on Guardian articles; and Cass Sculpture Foundation, which insert lists of articles from the Guardian about its sculptors into their biography and home pages.
Image taken from Roo Reynolds at Flickr.