I've been seeing a few people having trouble getting this working, so here's the quick run-through...
Why?
HTML5 Video has a smaller filesize, downloads much faster, is very cpu-friendly (i.e. it won't bog down the entire page), renders at a higher FPS and better quality (depends on the source material).
The Basics
You need to use the [h5loop] tag and put the url of the video inside. You can also use the [h5video] tag, but this will neither auto-play nor loop automatically.
The supported extensions / mimes are webm, obv and mp4 but you should generally use mp4 and never worry about the others.
Getting Video Address
Chrome
Right Click -> "Copy Video Address."
Firefox
Right Click -> "Copy Video Location."
Safari
Right Click -> "Copy Video Address."
Then just wrap in [h5loop] and you're done...
Result:
Getting Video Address Manually
Note: This is the hard way, if your browser is listed above you should be able to get the video address painlessly through the right click menu.
Embedding w/ Imgur
Result:
Embedding w/ Gfycat
You will need to try either zippy or giant. These are not interchangeable, if you see an ugly XML error try the other one.
Result:
Embedding w/ Anybody Else
If the link you have ends in .mp4 you can use it directly. Otherwise you'll need to look at the page source. Typically you'll either find this in the "View" or "Develop" menu. Otherwise try a quick google search for your browser / os combination.
Then you'll want to search for ".mp4" and a url address. In the following example you don't need to preceding two slashes in the url. Chrome allows me to click on the url and get the webm to open in a new tab.
Hosting Original Content
I recommend https://imgur.com since it's easy to use. /r/webm has some other recommendations as well if you need to upload very large videos / gifs.
When not to use?
FHC has a implementation that does not auto-repeat, so any video where looping is essential (cinemagraph, loop animation, etc.) should not use these.
Ethos
HTML 5 video should not be used as a replacement for .gif (i.e. in the GIF Thread) unless it is relatively short and has no sound (this is less of a concern w/ h5loop as it is auto-muted). If it doesn't meet those two criteria it probably belongs in the Video Thread.
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