Thank you again for explaining things further, however I still can't seem to make it all work - I must be doing something wrong and I can't find any instructions anywhere to help me further - hopefully you may be able to...
So, both my iPhone 4 and Apple TV are connected to the same network via the same wireless router in my house. I went into the Apple TV Settings and, under General, I made sure that the Airplay Settings was set to Show. I then connected with BBC iPlayer on my iPhone 4 and started to play a streaming video of a BBC programme. I then tap the iPhone screen and the whilst the content continues to play just on the iPhone, nothing else happens, no control bar appears, and so no Airplay icon either. If you can help further, I would be very grateful.