The Dropbox API currently on version 2 allows developers to build applications that work with a user's Dropbox content. It essentially gives you file system like end points such as copy, delete, get, list_folder, move, restore, search, and upload.
There are four types of authentication available. User authentication, team authentication, app authentication, and no authentication you can read more about these different methods on their documentation page
Dropbox has guides organized by language. They also have a well contributed cummunity SDK index
The overall status of Dropbox is available here
The standard terms of service for the Dropbox service applies to the API and can be found here https://www.dropbox.com/privacy#terms