Add a url to video file in your ReadMe.
Github now supports videos, see more detailed answer here: https://stackoverflow.com/a/4279746/1092815
You can see a live example here (See at the end of the readme):
https://github.com/alelievr/Mixture/blob/0.4.0/README.md
Github Pages
I strongly recommend placing the video in a project website created with GitHub Pages instead of the readme, like described in VonC's answer; it will be a lot better than any of these ideas. But if you need a quick fix just like I needed, here are some suggestions.
Use a gif
See aloisdg's answer, result is awesome, gifs are rendered on github's readme ;)
Use a video player picture
You could trick the user into thinking the video is on the readme page with a picture. It sounds like an ad trick, it's not perfect, but it works and it's funny ;).
Example:
[![Watch the video](https://i.imgur.com/vKb2F1B.png)](https://youtu.be/vt5fpE0bzSY)
Result:
Use youtube's preview picture
You can also use the picture generated by youtube for your video.
For youtube urls in the form of:
https://www.youtube.com/watch?v=<VIDEO ID>
https://youtu.be/<VIDEO URL>
The preview urls are in the form of:
https://img.youtube.com/vi/<VIDEO ID>/maxresdefault.jpg
https://img.youtube.com/vi/<VIDEO ID>/hqdefault.jpg
Example:
[![Watch the video](https://img.youtube.com/vi/T-D1KVIuvjA/maxresdefault.jpg)](https://youtu.be/T-D1KVIuvjA)
Result:
Use asciinema
If your use case is something that runs in a terminal, asciinema lets you record a terminal session and has nice markdown embedding.
Hit share button and copy the markdown snippet.
Example:
[![asciicast](https://asciinema.org/a/113463.png)](https://asciinema.org/a/113463)
Result:
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