The Dropbox API does enable you to build an app that users can connect their accounts to, in order to interact with their files. For example, once a user is linked to your app, the app can upload and download files to and from their account, as well as share files.
To get started, we recommend using one of the official SDKs, if possible:
https://www.dropbox.com/developers
They have documentation and tutorials to help you start your integration.
If we don't have an SDK that suits your needs though, you can use a third party library, or use the HTTP endpoints directly.
For example, to get a shared link to a file or folder, you can use the /sharing/create_shared_link_with_settings endpoint.