The Twitter connect enable you to login a user into your website using his Twitter account, and the ability to request and display his Twitter information into your application or website.
Once a user has granted your application the permission to interact with his Twitter account, you can:
- Store the Twitter token into your database to authenticate the user in the future
- Enable the user to update his Twitter status from your application
- Display to the user's Twitter followers
- etc...
The possibilities are large and depends on what you need to implement in your application.