AirPlay is the feature of iOS that displays the output of an iOS device or Mac computer to an AirPlay enabled device. This can mean redirecting the output of a presentation to an AppleTV or even some televisions now come with AirPlay as a feature. Reflection is an inexpensive app for Mac OS X and Windows that can reflect AirPlay output from an iOS device or a Mac to a Mac or Windows computer.
To enable AirPlay on a Mac, use the Displays tab on the Display System Preference pane to set the AirPlay option for “Show mirroring options in the menu bar when available”.
This will display an option in the menu bar, shown here as the icon on the far left.
Clicking on this icon shows every AirPlay enabled device that your computer can communicate with.
Simply click on the device to output your display to and the system will appear on that device, unless a password has been set. Out of the box, the Reflection app just works when opened. Click on the Reflection menu and click on Preferences to configure a password.
From the Preferences screen, unclick on the Use my Mac’s name in order to use a custom name for a system in the AirPlay Name field. Enter a password in the Password field in order to prompt users who attempt to direct their output to the system to enter a password. Click on Done.
We’ve shown how to connect to an AirPlay device from Mac OS X. To connect from iOS, doubletap on the home button and then use the bottom bar to scroll/swipe your way all the way to the left.
From this menu, tap on the video icon and then tap on the name of the system you wish to redirect output too. Then slide the Mirroring option to ON (if you don’t see a Mirroring option then check for a profile that disables the AirPlay option).
The device is then displayed on your computer. That’s it.
Also, there are other apps and a more modern version of Reflector (v2) covered in an article by JP Zhang at http://www.softwarehow.com/ios-screen-recorder-guide/.