I am trying to stream a DVD from my MacBook Air to my TV with the Apple TV. When it comes up on the screen, it only shows a checkerboard design. If I disengage the Apple TV, the DVD shows perfectly on the MacBook Air screen. Apple Support says it is because of copyright on the DVD that won't let it show on the TV. I don't know why it would be any different that showing it on a DVD player. Is there a way to work around this block to be able to stream a DVD?
Try using VLC to play the DVD instead of the OS X's DVD Player app.
You can also rip your DVD to a format the Apple TV can play directly using Handbrake, but obviously this isn't really an on-demand solution (it will probably yield better quality than VLC + AirPlay Mirroring though).
Also see this question for details on how you could set up direct streaming without AirPlay mirroring, which should yield better quality (at the cost of some time setting it up at first): Can I stream any video played with VLC player to Apple TV?