First, a brief explanation. I run a local Mac Users Group (MUG) and many of the members are at the beginning to intermediate level. When I'm asked the same question repeatedly at my MUG I'll create a quick tutorial on the subject and post it for the group. I understand the audience for this site is a bit more advanced and and may find some of these tips unnecessary. Nevertheless, the content will be useful to someone so I figured why not post it.
By the way, these tutorials are made with Clarify. Full disclosure, they're a future sponsor of Mac Power Users, but I've been using their software for years.
Enable AirPlay Mirroring on your iPhone or iPad
AirPlay mirroring is available on the iPhone 4s or iPad 2 and later devices and requires iOS 5 or later. With AirPlay mirroring you can display your iPhone or iPad's screen on an AirPlay capable device like an Apple TV.
First, let's take a moment and distinguish the difference between AirPlay mirroring and regular AirPlay. AirPlay has been around for a while and allowed you to project video or audio content from certain enabled Apps to comptable receivers. You may see the AirPlay icon in a YouTube video within Safari or anytime a QuickTime movie is played or when browsing videos. AirPlay mirroring is different in that it displays whatever the contents of your iOS device's screen. Why is this important? Sometimes you want to show your entire screen, like when giving a demo. Or some applications don't allow AirPlay streaming (such as the ABC and NBC video Apps) so AirPlay mirroring is away to get around this limitation.
Note that certain applications (such as Amazon's video player) may specifically disable AirPlay mirroring since they don't want you to be able to broadcast their content to another device.
Engage the Multi-Tasking Bar
Double-click the home button to display the multi-tasking bar.
Swipe Right Twice
Swipe from left to right twice to navigate to the AirPlay menu
Select the Device
Select the device you want to AirPlay to. If you want to mirror your iOS display, turn "on" the option for mirroring. Note that devices with a speaker icon are only capable of broadcasting audio while devices with a display icon can broadcast video.
AirPlay Status Bar
When AirPlay mirroring is active, you'll see a blue status bar across the top of your device with the AirPlay icon.
Turn Off Mirroring
To turn off mirroring, repeat steps 1 and 2 and slide the mirroring optino to "off" or select your iPhone as the display.