Update: Everything below is still correct, but the official YouTube API for Android is now available.
By far, the easiest way to play a YouTube video on Android is to simply fire an Intent to launch the native Android YouTube app. Of course, this will fail if you are not on a certified Google device, that doesn’t have the complement of Google apps. (The Kindle Fire is probably the biggest example of such a device). The problem with this approach is that the user will not automatically wind up back at your app when the video finishes; they have to press the Back button, and at this point you’ve probably lost them.
As a second option, you can use the MediaPlayer API to play YouTube videos. But there are three caveats with this approach:
1) You need to make a call to YouTube’s GData webservice API, passing it the ID of the video. You’ll get back a ton of metadata, along with it the RTSP URL that you should pass to MediaPlayer to play back an H.264-encoded stream. This is probably the reason why your attempt to use MediaPlayer failed; you probably weren’t using the correct URL to stream.
2) The GData/MediaPlayer approach will only play back low-resolution content (176×144 or similar). This is a deliberate decision on the part of YouTube, to prevent theft of content. Of course, this doesn’t provide a very satisfactory experience. There are back-door hacks to get higher resolution streams, but they aren’t supported on all releases of Android and using them is a violation of YouTube’s terms of service.
3) The RTSP streams can be blocked by some internal networks/firewalls, so this approach may not work for all users.
The third option is to embed a WebView in your application. There two approaches you can take here:
1) You can embed a Flash object and run the standard desktop Flash player for YouTube. You can even use the Javascript API to control the player, and relay events back to the native Android app. This approach works well, but unfortunately Flash is being deprecated on the Android platform, and will not work for Android 4.1 and later.
2) You can embed a <video> tag to play YouTube via HTML5. Support for this varies between various releases of Android. It works well on Android 4.0 and later; earlier releases have somewhat spotty HTML5 <video> support. So, depending upon what releases of Android your application must support, you can take a hybrid approach of embedding HTML5 on Android 4.x or later, and Flash for all earlier versions of Android.
There are several threads here on StackOverflow about using HTML5 to play YouTube video; none of them really describe the entire process you must follow in one place. Here’s links to a few of them:
Android – How to play Youtube video in WebView?
How to embed a YouTube clip in a WebView on Android
Play Youtube HTML5 embedded Video in Android WebView
All of this will get dramatically easier in the weeks/months to come; at Google I/O 2012, they presented/demoed a new YouTube API for Android that will support direct embedding of YouTube content in your application, with full support back to Android 2.2 (about 95% of the Android userbase as of this writing). It can’t arrive fast enough.