Twitter isn't really known as a platform for video, starting out as a text-only social site. Most people attach photos to tweets these days, including animated gifs which can look like videos. The main source of video on Twitter is - of course - Vine, the short video sharing site that Twitter launched back at the start of 2013. See also: How to download YouTube videos
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 Twitter
When 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.
If 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.
YouTube 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
HTML5 - Webm videos
These can be trickier to download as you may not be able to use a website to process the video and offer a download link. Various applications claim to download all streaming video online, but most have a free trial so you can test them out before paying.