There are 2 types of Twitter videos: animated GIFs and real videos. This post is about animated GIFs. They have the text "GIF" printed on them when they are not playing.
To download animated GIFs there doesn't seem to be an easy way in Google Chrome unless you use an extension.
- open the tweet
- right click on the video
- choose "This Frame" -> "Page Info"
- Under "Media" choose the mp4-file and click "Save As..."