There are many different methods to copy passwords stored in KeePass to other applications:
Context-Sensitive Entry List
Depending on which field you double-click in the entry list (main window), different actions are performed:
You can drag&drop all fields of KeePass entries into other windows:
Auto-Type is a powerful feature that sends simulated keypresses to other applications.
You can find more details about it here: Auto-Type documentation page.
KeeForm and other Plugins
There are a lot of plugins available integrating KeePass directly with other applications. For example, KeeForm (for Internet Explorer and Mozilla Firefox) allows completely automatic filling of webforms.
You can find these integration plugins on the KeePass Plugins page.