Since YouTube videos are mainly FLV (Flash Video) files, and Adobe Flash is unsupported on certain Cruz tablets (R100 series, T100 series, T301), the YouTube website will not play videos from within the browser app. See also for more info: Will it work on my Cruz?
A 3rd party app like the ones mentioned here can provide some YouTube video playback functionality.
TubeMate is a video download app available on the Cruz Market, perfect for R100 series.
The YouTube app is a limited streaming video playback app available on the Cruz Market for all other Cruz tablets.
Skyfire is browser app that allows for some Flash video download and playback functionality.