Web APIs are sexier that desktop APIs. Having an API allows the external developers to access your data or your services in a smart way. They can they use your data or services in ways you would not imagine.
Another popular category of APIs is Search. Search APIs can provide cache access, spell-checking, content analysis and much more. Amazon's API offers search on prices, images, customer reviews and affiliate sales.
A good resource for Web APIs is ProgrammableWeb, which acts as an encyclopedia of available APIs and how people use them.
Mashups are novel UI that enhance your data, e. g. by combining your local data with mapping information. Chicagocrime.org – one of the first map mashups was built prior to Google’s API being made public. It’s not all about maps – TagTV, Viral Video Chart, BlueOrganizer, Salesforce Adwords.
In the future we can see ContextAgnosticXmlHttpRequest, enhanced JSON – JSONRequest. Web APIs are all about work we do not have to do. So open your data, offer an API, let the others do the work!