If nothing happens when you toggle the Mirroring switch under your computer’s name in the AirPlay menu, this means there is a block in communication between the two devices. It's likely that a firewall on your computer or antivirus software is causing the block.   

It's also possible that the firewall on your router is preventing communication.

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

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

Corporate networks often block a number of ports from communicating across networks. You may need to have your 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 and mDNS in Multicast mode. Unless special steps have been taken, Reflector will likely not work across subnets or VLANs.

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

You can also find additional Windows firewall information here: http://windows.microsoft.com/en-us/windows-8/windows-firewall-from-start-to-finish

If all of this checks out, it’s possible that you’re running into an iOS 8 bug. Try giving your iDevice a hard reboot.