Admin embed Movie
Should embed any movie but needs full URL with https:// prefix trimmed and manually selected from pulldown and width and height to be added. Doesn't appear to work for Youtube...
Admin Embed Youtube
can use ID but better to use full URL
Youtube iFrame embed
according to https://developers.google.com/youtube/player_parameters#Embedding_a_Player this is the recommended method and the only one that appears to work on all devices.
The same principles will work for Vimeo and any other provider.
Everything needs to be set manually in HTML view, so only recommended for authors with advanced skills. There are lots of options (see link above) but a good starter for ten is as below with editable bits in bold. Set autoplay to 1 to have the video start automatically on pageload and autohide to 0 to keep the controls visible throughout. 390x640 is a good standard size (bearing in mind you can make it go full screen) but tweak these according to your site design/breakpoints.
<iframe id="ytplayer" type="text/html" src="http://www.youtube.com/embed/-L1Sni3eLDY?autoplay=0&autohide=1&origin=http://www.peakdistrict.gov.uk" frameborder="0" height="390" width="640"></iframe>