You do not need an internet connection, but you will need some kind of home network to get the media from your Mac to the Apple TV. The home network can be WiFi or Ethernet. You will not be able to stream movies from the iTunes Store or use any of the other Apple TV apps that need access to the internet for content, but you will be able to listen to your music and watch your movies and photos, as long as those things are already in your computer’s iTunes library.
AirPlay will also work fine as long as all the devices involved are on the same home network.
In other words, anything that you already have in your home will be accessible from the Apple TV, but anything that normally downloads from the internet would be off-limits.
