YouTube videos are encoded in FLV (flash video) format. Hence, you need a special FLV video player to watch the downloaded videos - they won't play on your regular Windows Media player, at least not without a plugin. One of the earliest and most popular FLV player (4.77MB) was created by Martijn Devisser.

There is also a free FLV video converter (4.73MB) by Riva that lets you easily convert from FLV to wmv, avi, mpeg, etc. so that you can watch them on your other devices.