Upgrading to Mountain Lion
To upgrade to Mountain Lion you must have Snow Leopard 10.6.8 or Lion installed. Purchase and download Mountain Lion from the App Store. Sign in using your Apple ID. Mountain Lion is $19.99 plus tax. The file is quite large, over 4 GBs, so allow some time to download. It would be preferable to use Ethernet because it is nearly four times faster than wireless.
OS X Mountain Lion - System Requirements
Macs that can be upgraded to OS X Mountain Lion
1. iMac (Mid 2007 or newer) - Model Identifier 7,1 or later
2. MacBook (Late 2008 Aluminum, or Early 2009 or newer) - Model Identifier 5,1 or later
3. MacBook Pro (Mid/Late 2007 or newer) - Model Identifier 3,1 or later
4. MacBook Air (Late 2008 or newer) - Model Identifier 2,1 or later
5. Mac mini (Early 2009 or newer) - Model Identifier 3,1 or later
6. Mac Pro (Early 2008 or newer) - Model Identifier 3,1 or later
7. Xserve (Early 2009) - Model Identifier 3,1 or later
To find the model identifier open System Profiler in the Utilities folder. It's displayed in the panel on the right.
Are my applications compatible?
See App Compatibility Table - RoaringApps.
About AirPlay and Airplay Mirroring
AirPlay Mirroring requires a second-generation Apple TV or later, and is supported on the following Mac models: iMac (Mid 2011 or newer), Mac mini (Mid 2011 or newer), MacBook Air (Mid 2011 or newer), and MacBook Pro (Early 2011 or newer). For non-qualifying Macs you can try using Air Parrot.
Several Apple Articles Regarding AirPlay
Apple TV (2nd and 3rd gen)- How to use AirPlay Mirroring
How to set up and configure AirPort Express for AirPlay and iTunes
About AirPlay Mirroring in OS X Mountain Lion
iTunes 10- About playing music with AirPlay
Troubleshooting AirPlay and AirPlay Mirroring
Using AirPlay
Thanks to the $15 Beamer, AirPlay streaming is still possible on Macs that do not support Airplay and mirroring.
Another solution is the Air Parrot.