The latest Twitter update provides native support for animated GIFs on Twiiter.com, as well as Android phones and iPhones. In the past users had to use third-party platform Giphy in order to display animated GIFs in their Twitter posts. But with the Twitter update, users can now upload GIFs directly from their devices. However it seems that the clips won't automatically play (unlike the video clips in Instagram), so you will have to press the play button before they play.
Source: Twitter Support