Google has updated the YouTube app for Android, but if you have a custom rom like CyanogenMod installed you may find that the install is failing from the market. This can be easily fixed by renaming (or deleting) the old YouTube app before trying to install the new one. This post assumes you have rooted your device, seeing how that is necessary to load the custom ROM anyway.

Open a terminal emulator and become root. Mount the /system filesystem as read/write using mount -o rw,remount /system

Change into /system/app

Rename (or remove, if you are brave) YouTube.apk

Reboot. One the phone comes back up, you should be able to download and install from the market like normal.