Q: I am interested in AppleTv.  Must I have WiFi; how does mirroring work?

You can connect an Apple TV to your network via Ethernet or WiFi.

Here are the tech specs for the ATV. Not necessarily wifi but you do need an Internet connection in order to stream online media, download apps/games...

http://www.apple.com/tv/specs/

As for home share and AirPlay/mirror, both devices need to be on the same network but there is a peer to peer option that would not require network...

Use Home Sharing to share iTunes content with other devices - Apple Support

Get help with AirPlay and AirPlay Mirroring on your iPhone, iPad, or iPod touch - Apple Support

First, you need an internet connection to the Apple TV to use it and a TV with an HDMI input.  The internet connection can be via an ethernet cable or via WiFi.

If your internet is connected via WiFi, and you have a iPhone, iPad, or IMac connected to the same WiFi, the Apple TV offers a bonus feature where you can view the screen of the iPhone, iPad, or iMac on your TV.

This is handy for viewing photos, videos, or playing games on your iPhone, iPad, or iMac while viewing it on a large screen and is termed mirroring or as Apple terms it AirPlay.