why doesnt the youtube and imessege work on my ipad

What are you doing? What exactly is not working?

What happens when you use them ? Do you get any error messages, are the apps frozen ... ?

Have you tried closing the two apps completely and then doing a reset and seeing if they work after the iPad has restarted ? To close them : from the home screen (i.e. not with YouTube or Messages 'open' on-screen) double-click the home button to bring up the taskbar, then press and hold any of the apps on the taskbar for a couple of seconds or so until they start shaking, then press the '-' in the top left of the two apps to close them, and touch any part of the screen above the taskbar so as to stop the shaking and close the taskbar.

To do a reset : press and hold both the sleep and home buttons for about 10 to 15 seconds (ignore the red slider), after which the Apple logo should appear - you won't lose any content, it's the iPad equivalent of a reboot.

Do other internet apps work? For example Safari?

For iMessages there is some troubleshooting on this page : http://support.apple.com/kb/TS2755

Has the YouTube app previously worked on the iPad, and have you ever synced the iPad with your computer ? This page for YourTube (which refers to a "Cannot connect to YouTube" error message) says :

To use certain services, iTunes must download and install an authentication token on the device. Because iPad and iPod touch do not require an active Internet connection to activate, these tokens may not be downloaded and installed during initial setup.

You can resolve this issue by verifying that iTunes has an active Internet connection. Use the following steps:

1. Open iTunes on your Mac or PC.
2. Verify that your Mac or PC has an active Internet connection by accessing the iTunes Store in iTunes.
3. Connect your iPad or iPod touch to your Mac or PC.
4. After the sync completes, verify you can access YouTube.

Has anything changed with your router e.g. updated firmware which might be blocking them ? And have you tried re-booting your router and/or resetting your network connections on the iPad via Settings > General > Reset > Reset Network Settings and re-connecting to your network ?