If the AirPlay mirror icon isn't displaying on your iOS device, it isn't able to see the computer running Reflector 3 for one reason or another.


If you're on a home network, the firewall on your computer may be preventing the devices from communicating. It's also possible that the firewall on your router is preventing internal network communication.


Microsoft has included instructions for allowing applications through the Windows firewall here:

http://windows.microsoft.com/en-US/windows7/Allow-a-program-to-communicate-through-Windows-Firewall


If you're using a Mac, Apple has provided similar instructions here:

http://support.apple.com/kb/ht1810


Corporate and school networks often block a number of ports from communicating across networks. You may need to have the school's IT department allow TCP ports 7000, 47000, 7100, 49228, 50259 and UDP 62572, 54780 to be open to communicate. The network must be also allowed to run Bonjour, MultiCast, and mDNS.


You must also be on the same VLAN or subnet within the network to connect, Apple's Airplay cannot cross subnets/VLANs, this is a limitation of Apple's AirPlay. 


Apple has provided more instructions on this issue here: http://support.apple.com/kb/TS4215