However, you can link to pretty much any online video in a tweet, so here's how to download those videos.
How to download video from TwitterWhen you see a video you like in a tweet, you won't be able to download it directly. So, first figure out the source of the video, either by clicking on the link in the tweet or using your web browser's developer mode (usually accessed by pressing F12) to see the source.
Videos can be embedded in many ways, including from YouTube, Vimeo, Vine and native HTML 5 video players.
The way to download the video will depend on its type.
VineIf it's on Vine, for example, just click through to the Vine website and right-click on the video in your browser and choose Save video… (or a similar option). The file will then be stored on your hard drive as an MP4 file which most video players can play.
YouTubeYouTube videos are easy to download, too. Lots of websites allow you to paste in the link to the video and then give you options to download the video in various qualities and formats. Examples include savetube and keepvid.
There are also applications which can do the same job. See: How to download YouTube videos for a step by step tutorial