I have mountain lion and Apple TV. iTunes only displays the option for AirPlay. But it doesn't display on the OSX top bar.
I have a MacBook Pro 2.66 GHz Intel Core 2 Duo
Airplay Mirroring in 10.8 is only compatible with machines that have a Core i-series processor. Your Core 2 machine is not on the supported hardware list for Airplay Mirroring with 10.8.
You should check out AirParrot, which is $9.99, but can do AirPlay Mirroring on older hardware (and OSes) as well.
There are certain requirements for using airplay in ML. Here is a list of all supported Apple Computers:
Depending on the Year and Model of your machine, you may not be able to use AirPlay mirroring.
If your computer is not listed as being compatible, then it will not work.
You can alternatively use a program called air parrot which will do roughly the same thing, just not as nicely.
The reason has to do with the graphics cards that the machines use, as explained here.
I hope that this answer helps you out.
Airplay mirroring is for full system side mirroring of the whole OS, not limited to just a video, or slideshow etc. As such, you computer (like mine) may well be able to use airplay within iTunes, whilst not being compatible for mirroring the whole display.