You have similar options for streaming audio from iOS devices. To stream all of your device's audio, double-press the Home button to access the task switcher, and then swipe to the right until you reach the volume slider; next to the slider is the iOS AirPlay button. Tap it, and then tap the desired AirPlay destination.
To stream audio from just a specific app, that app must provide an AirPlay-selection button within the app. If it does, just tap it to see the same list of all AirPlay receivers on your local network; tap one to stream to it. Note that unlike on a Mac, iOS can stream to only a single AirPlay destination at a time.
Using AirPlay for video
Streaming video from iOS to a recent Apple TV works much like streaming audio. To mirror a recent iOS device's screen on your TV, again press the Home button to access the task switcher, and swipe to get to the AirPlay button. Tap the button, choose the desired Apple TV, and switch the Mirroring option to On.
To stream from an individual app that supports AirPlay video, just find and tap the AirPlay button in the app, and then choose your Apple TV from the list that appears. (I can't show you the procedure here, as I can't record my iPad's screen without mirroring it.)
Streaming video from iTunes on a Mac is similarly simple: Just click the AirPlay button in the video-playback window, and choose an Apple TV.
If you want to stream video that's not in iTunes, you can use the third-party utility Beamer: Choose your Apple TV as the destination, drop any supported video file onto Beamer, and you're done.
Mirroring all your Mac's video on your TV—meaning anything and everything on your Mac's screen shows up on your TV, as well—works only under OS X 10.8 Mountain Lion on very recent Macs. There are two ways to configure AirPlay mirroring: in the Displays pane of System Preferences or using the AirPlay Mirroring menu. The latter shows up whenever OS X detects a compatible Apple TV on your local network: Choose your Apple TV to start mirroring. You can also tweak the output resolution to choose the best for your TV or the best for your Mac.
(If you don't have a Mac that supports AirPlay mirroring the third-party utility AirParrot gives you similar functionality.)
Stopping the stream
Whichever type of streaming you're doing, with whichever device, you can stop streaming by using the same AirPlay control or menu through which you originally started streaming. Alternatively, if you're streaming from a particular app, quitting that app usually stops streaming. When streaming to an Apple TV, you can also stop AirPlay streaming by pressing the Menu button on the Apple TV's remote.
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