Hi.
Without stating the obvious, this as been a problem for some time, after trawling for answers for several days after not being able to get my ATV3 working straight out of the box with AirPlay, I have come up with some solutions to some but not all of the problems raised.
1 there have been several updates to IPads, Macs & ATVs. Make sure you have the latest and greatest on all devices.
2 make sure that the apps that you are using support AirPlay, if they do not you will see a music symbol on the ATVS.
See the list on this page for some of the apps that work. This is not the full list, TVCatchup now works over airplay.
http://theapple.tv/apps/list-of-airplay-enabled-apps/
3 if you are not using an apple router then reduce the security on your NAT firewall in the router settings
3 this may be a little patronising to some of you but make sure your device is set to Apple TV.
If you are getting an HDCP error on the screen you will require a new HDMI cable, preferably a good quality one.
If all else failed backup and restore your devices