API:Categorymembers

List of pages that belong to a given category, ordered by page sort title.

Example
If you're using "cmcontinue" parameter for pages, the "cmtype=page" parameter is optional, but for subcategories (or files), the "cmtype=subcat" (or "cmtype=file") is necessary.

Retrieving via AJAX
Below is sample code for retrieving the category list via an Ajax request: