How do I send a YouTube video from my desktop to my Android tablet?
The Chrome to Phone extension is what you'll be wanting.

To get Chrome to Phone, install the Chrome to Phone extension in your Google Chrome browser and the Android app on your phone, and you’re ready to go. 

Chrome to Phone extension adds a button to your Google Chrome browser that instantly sends the current YouTube video, web page, map, or selected phone number or text to your Android device running Froyo (or Android 2.2) or up.

For Firefox users, there's a compatible Firefox addon FoxToPhone, which sends URLs etc. from Firefox on the desktop. It still needs the same app installed on the phone.