An excerpt from http://www.sitepoint.com/wordpress-categories-api/ , by Jérémy Heleine
In a previous article we saw how to easily work with categories in WordPress. We covered how we can sort our posts into different categories, which can then be edited or deleted.
Having covered the basics, it’s now time to have a look at something that will be of more interest to developers: the Categories API, and how to retrieve and display category data.
Like the Links Manager API, the Categories API is a big topic, even bigger in terms of the number of functions available. In this article, we’ll be covering one function that is useful when we want to list our categories.