So, you want to connect your Apple TV to your laptop? Cool! You've got Mountain Lion, you setup everything (sso you think), and you go to do it, and it dosen't work! What do you do? Read this guide to SOS.
Did you setup Home Sharing?
Home Sharing is required to pair an Apple TV with a Mac.
Did you connect both devices to the same Wi-Fi Network?
AirPlay thrives on WiFi, so without it you can't do anything. If you somehow don't have WiFi:
On your Mac, move your mouse to the WiFi icon in the status bar. Select "Create a network". After that, connect your iPad to the same network, and in theory it should work.
Is your Apple TV on and connected?
Again, make sure it is connected.
Still not working?
Turn off and on your Mac and your Apple TV.
Disconnect and reconnect from the WiFi on both devices.
Ensure Home Sharing is enabled on both devices.
If further **** ensues, call Apple Support, who can help you more.