The Google Maps API is listed as the most popular API in the ProgrammableWeb API directory. Because of its ease of use, plus the infinite possibilities for hacking together mashups, has made it the most popular API by far, way beyond the Twitter API and Youtube API.
Quick, what is your House of Representatives district? Many people don't know this number, or the member who represents them. The New York Times' new Districts API makes it easy for your application to answer the question for any point within New York City, though it could serve as a template for a nationwide--or even international--service.