Welcome to the Apple Community.
Try the following steps, check whether things are working after each step where appropriate, before trying the next.
- Check AirPlay is turned on on the Apple TV (turn it off and on if it already is)
- Check that both devices are on the same network (Settings > Wifi, on the mobile device and Settings > General > Network, on the Apple TV).
- Restart the Apple TV (Settings > General > Restart).
- Restart the Apple TV by removing ALL the cables for 30 seconds.
- Restart your router.
- Restart your mobile device.
does the remote app works?
does the ios devices see the computers shared itune lib?
does the appletv see the computers shared itune lib?
can the ios device use the appletv as airplay audio speakers?
can the computer use the appletv as airplay audio speakers?
is the appletv connect by wifi ?
make sure the ios devices is not connecting using 3g to the internet as airplay does not work across the internet
check the wifi routers settings if ports required for airplay are blocked by the router