One of the larger annoyances of YouTube video producers is that they may restrict their video to Flash only. Even if you turn on HTML5 support, videos do not play. Here is a quick fix for some of those types videos if you use Safari.
The basic idea is that you change the user agent to "Safari for iPad". This makes YouTube think that you are browsing from an iPad, and it will serve up a non-Flash video.
First, you have to turn on the Develop menu. Go to the Advanced tab of Safari Preferences. At the bottom, check on “Show Develop menu in menu bar”. You will now have a Develop menu (between Bookmarks and Window).
Go to that menu, and pick User Agent. A submenu will show various browser types. Select “Safari iOS 4.3.3 – iPad”. The YouTube page will immediately reload, and in most cases, you will be able to watch the video.
I haven’t tried this trick in Chrome, yet, but will update once I do.
UPDATE: This also works to play music without Flash on Tumblr, except that you will have to select Safari iOS for iPhone.
UPDATE 2: Well, YouTube seems to have started blocking this work-around for Safari. It does seem to still work in Chrome: use the User-Agent Switcher extension.