With the ZT180 tablets access to the Android Market can sometimes be a bit sketchy. Unfortunately that's one of many knockoff tablets on the market and it doesn't "officially" support access to the Android Market. The market is there due to the app being fooled into thinking that your tablet is actually something else.
For these reason you first need to check to make sure that the market even works. Once verified you can do what Steven recommended and search the market for the Youtube app. Once you find it, tell the market app to download and install it on your tablet.