What is an accessory shoe?

The Basics:

An Accessory Shoe is the small metal bracket or mount on top of a camera used to attach a Speedlite Flash, and other accessories such as microphones.

A Little More Detail:

An Accessory shoe can be referred to as ‘hot’, whereby it communicates electronically with the attached accessory, or ‘cold’ where it doesn’t.   

For example, all Accessory Shoes on Canon EOS cameras are Hot Shoes, which enable the triggering of a Speedlite Flash. An example of a Cold Shoe is where you can mount a Speedlite off-camera using a special bracket, with the electrical communication taken care of with an off camera cable.
See Hot Shoe for more information.
OTHER WORDS UNDER A