HTML5 solution with simple fallback.
HTML5 support is actually really good by now. (http://caniuse.com/#feat=mpeg4)

But lets use a polyfill to support old browsers:
(sources: http://html5please.com, https://github.com/Modernizr/Modernizr/wiki/HTML5-Cross-browser-Polyfills)
Two good candidates:
https://github.com/johndyer/mediaelement (Used in wordpress core.)
http://www.videojs.com/

Ideally video component dialog will show dam chooser filtered to show videos. But this is not a strict requirement.

Context: this component starts in the demo but will eventually be in a separate library of components - either in MTE or another "Extra Components" module.

Usage in Demo:
Add a video element to the company page after the jumbotron, but before the text. (https://demopublic.magnolia-cms.com/about/company.html)
Video of grand canyon is attached.

Video should have caption: "The holiday of a lifetime. Helicopter tour over the Grand Canyon."

Video should have 100% width and undefined height so that it will scale responsively.
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Note on rights - video is public domain from the US National Park Service. See: http://www.nps.gov/grca/learn/photosmultimedia/b-roll_hd_index.htm
Note on video preparation: - Quicktime can be used to export trimmed clips. Free "Hand Brake" program for osx can be used to resize and export in different formats.