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Assuming both devices are on the same network and that AirPlay is not turned off on the Apple TV, then simply tap on the screen when you are watching content you wish to stream to your Apple TV, then tap the airplay icon that appears in the control bar, choose the Apple TV from the menu that appears.
When displaying the content you wish to mirror on the iPad 2 (or better), iPad Mini, iPhone 4S (or better), double tap the home button (quickly) and swipe the bottom row of apps to the right to reveal the playback controls, tap the AirPlay icon and select your Apple TV from the list of available devices.
If the problem continues, try restarting the Apple TV by removing ALL the cables for 30 seconds, or resetting it using the reset option under general. You should also try restarting your router and iPad.
It answers both questions!
AirPlay to an Apple TV, from an iPad Mini, can be done as either "whole screen mirroring", or from "AirPlay-enabled apps". The built in apps "Videos" and "Music" are two examples.
Re-read Winston's reply, the paragraph beginning with "When displaying..." tells how to turn mirroring on.