AppleTV Air Mirroring: does the signal go through the WIFI router?
Two different AirPlay devices exist:


AirPlay transmitters (such as iPhone/iPad/iPod/iTunes)
AirPlay receivers (Apple TV, AirPort Express and audio system with built-in AirPlay)


Apple TV will connect to your WiFi modem to get Internet access.

It will also connect to your computer (without modem) for Airplay/mirroring.

It could also connect true the AirPort Express modem if you have it set up for that. Check what settings do you have in this article.