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. (Also try removing it’s power cord for at least 30 seconds)
- Restart your mobile device.
The following article(s) may help you.
Troubleshooting AirPlay
Troubleshooting Wi-Fi networks and connections
Recommended Wi-Fi settings
Wifi Diagnostic Software (for Mac users)
You may also find some help on this page, where I’ve collected some of the more unusual solutions to network issues.