Flash is all but gone from the mobile world, with Adobe announcing it won't be supporting Android 4.1 and future versions from now on, and with Google releasing Chrome for ICS and JellyBean without any kind of support for Flash. This however has never been a major problem, at least for Youtube, because from day one Google made it so any Youtube video could be played by the native Youtube application.
Lately Google has been redirecting users to the mobile site version of Youtube when they hit play on an Youtube app, and that may become the standard for the browser in the future, but Google will also allow developers to embed videos within their native Android apps from now on as well. This means that you will be able to play Youtube directly in someone else's app.
According to Google this new API will be very easy to use, as it takes only 3 lines of code, and it gets support back to Froyo, which means at least 95% of the Android devices out there. Google previewed this functionality and a lot more at Google I/O 2012 in their Youtube session for developers: