AirPlay Mirroring on mid 2010 MacBook Pro

AirPlay Mirroring in OS X Mountain Lion takes advantage of the hardware video encoding capabilities of 2nd and 3rd Generation Intel Core processors. Hardware video encoding allows AirPlay Mirroring to efficiently deliver high frame rates while maintaining optimal system performance.

You will not be able to use AirPlay mirroring due to hardware limitations. However, you can still steam music, movies, & photo's to the Apple TV.

Also if you have the new iPod 5th generation, or iPhone you can stream content from the iDevice to the AppleTV as well:

http://www.youtube.com/watch?v=u5RfCh6jpBM

Here's a little more info:

So why can’t older Macs support AirPlay Mirroring?

It comes down to hardware advantages of mid-2011 Macs, as they received Sandy Bridge i5/i7 chipsets that included separate dedicated hardware encoding for H.264. AppleInsiderexplains why older Macs can support AirPlay, but not AirPlay Mirroring:

These systems are the oldest machines capable of supporting AirPlay Mirroring because they are the first to deliver dedicated hardware encoding for H.264. Without a CPU capable of crunching this task using specialized hardware, earlier Macs simply can’t transmit video fast enough without a lot of heat and screaming fans.

Older Macs can already send video from iTunes to Apple TV via AirPlay, but they stop local playback while doing this. Mirroring requires the system to produce two video images, one driving the local display and one to be wirelessly delivered to the external screen.

AirPlay doesn’t just relay video from the computer (or iOS device) to Apple TV. It scales down the video to fit on an HDTV resolution, and has to convert the colorspace from the computer’s RGB to the native YUV that televisions use. Both tasks require a lot of processing resources, so without the extra hardware available on newer Macs, mirroring isn’t possible.

Users have noted the app AirParrot mimics AirPlay Mirroring on pre-2011 Macs with both video and audio. However, note that live video encoding is a hardware intensive task–so note your Mac will run very hot with the fans on full blast.