Is there a Wikipedia API?

MediaWiki's API is running on Wikipedia (docs). You can also use the Special:Export feature to dump data and parse it yourself.

More information.

Wikipedia is built on MediaWiki, and here's the MediaWiki API.

If you want to extract structured data from Wikipedia, you may consider using DbPedia http://dbpedia.org/

It provides means to query data using given criteria using SPARQL and returns data from parsed Wikipedia infobox templates

Here is a quick example how it could be done in .NET http://www.kozlenko.info/blog/2010/07/20/executing-sparql-query-on-wikipedia-in-net/

There are some SPARQL libraries available for multiple platforms to make queries easier