HTML5 Video works everywhere except the iPad / iPhone / iOS devices.
Double check to make sure your video is H.264 encoded. Most other browsers will play other formats, even Safari on OS X via the Quicktime plugin. There are exceptions (notably browsers that require Vorbis), however, most mobile devices at this time require that specific codec.
HTML5 Video plays correctly on your local copy of a site you’re building, but once you upload it to a server, it breaks.
This can happen if the server sends the video back with the wrong MIME-Type, usually text/plain or something similar. What you need to have the server send the correct MIME-Type. If you’re using Apache, add these lines to your .htaccess or server configuration file:
AddType video/ogg .ogv
AddType video/mp4 .mp4