How to fix Apple TV Wi-Fi connection issues on iPad?

Just had an incident where Apple TV dropped its Wi-Fi connection. In the end, restarting fixed it, but along the way…

When I went to Network in Settings, it said Wi-Fi was not connected, so I selected that, but I did not get a list of available networks, just “other” as an option to enter a network name. The interesting thing was that I could use my iPad's virtual keyboard to type in the network name on the Apple TV. (The ATV didn't recognize the name which was entered correctly, so at that point I backed out and restarted.) BUT, I always thought the iPad uses Wi-Fi to connect to the ATV, and since there was no Wi-Fi connection on the ATV, how could the iPad communicate with it? Is this connection over Bluetooth? Ultimately it’s not really important, but I am curious.

[Re-Titled by Moderator]

Original Title: Confused about iPad-Apple TV wireless connection