If iPhones/iPads don't support Flash, how do they play YouTube?
They use a format called H.264. YouTube and other video sites deliver content in this format also (so that their videos are viewable on iPad/iPod touch).

One advantage of H.264 is that hardware based encoding/decoding can be used.

See also Steve Jobs's Thoughts on Flash

H.264 is one of the formats for video support in the latest version of HTML, HTML 5. Apple supports this in Safari and Google in Chrome*.

*Google recently announced that it will no longer support H.264 in the future versions of Chrome.

Because of this H.264 is kind of hot of late. See these articles.