you need an appleTV
there are 2 modes of video airplay.
1 - the video plays on the screen, for apps like the native video player, youtube, and some games. you will get a fullscreen image, with no user interface overlayed.
some games will give you a "second screen" using this, the controls will be on your phone, while the main action display will be on the TV.
2 - mirroring, the display is copied directly to the screen, What you see on your device is what you see on the TV, top bar with clock, any buttons and such.
Due to aspect ratio differences, you will have black bars at the sides of your television screen, and it might not fill the screen top to bottom either. also if your device is in portrait or landscape will affect the display on the TV.
a lot of media apps (things like cartoon network, or the olympics) use this mode, as the content providers see it as possibly interfering with their revenue stream if you could watch the video in full screen on your TV.
it depends on the app developer as to weither the app supports full screen video airplay or not.
you must have a iPhone 4S or iPad 2 or newer to do mirroring.
I'm not sure if jailbreaking a 4 will allow you to do it or not, i'm guessing not, as i believe the 4S has a special chip to help with the video encoding. in much the same you you mac must be less than a year old to do airplay mirroring.