The service is the work of a separate company, Vine, which was acquired by Twitter before release. Vine can also be used independently of the social network but, for now at least, it's an exclusively mobile service.
According to Twitter, it's currently working to bring Vine to other platforms, with Android presumably first and foremost among those.
You can see the first tweet sent using the service, by Twitter CEO Dick Costolo, below.
Steak tartare in six seconds. vine.co/v/bOIqn6rLeIDvia @dhof
— dick costolo (@dickc) January 23, 2013