Many questions can be answered here, and definitely a useful stop before heading over to the user forum to ask:
The files of WordPress define many useful PHP functions. Some of the functions, known as Template Tags, are defined especially for use in WordPress Themes. There are also some functions related to actions and filters (the Plugin API), which are therefore used primarily for developing Plugins. The rest are used to create the core WordPress functionality.
Many of the core WordPress functions are useful to Plugin and Theme developers. So, this article lists most of the core functions, excluding Template Tags. At the bottom of the page, there is a section listing other resources for finding information about WordPress functions. In addition to this information, the WordPress phpdoc site details all the WordPress functions by version since 2.6.1.