WordPress.org

How do I create a function?

See Installation documentation

How do I use this with my app?

The example is for flash but the API can be used with just about anything from mobile apps to ajax/jquery javascripts
See the flash example: /wp-content/plugins/flash_api/flash_example/flash_api_example.fla

Can I set API keys for individual Users?

Yes!
1. Edit the user's profile -> Flash API -> API Domain value.
2. Edit the user's profile -> Flash API -> API API Key value.
2. Give that apiKey to the user to use for their flash apps.
3. The user must specify the domain parameter when calling functions from mobile apps or other domains. By default, the plugin will check the $_SERVER["HTTP_HOST"] value for the correct domain.

You can then alter your services to work only for specific API keys by checking the $_REQUEST["apiKey"] value.
You can also set user permissions within your services via WordPress's user levels manangement (see http://codex.wordpress.org/User_Levels).