Typically a turkey shoot is conducted using shotguns at a specified range with the resulting shot patterns on a paper target determining the winner. In most cases the contestants will all use the same ammunition (so a 12-ga. competition, a 16-ga. competition, a 20-ga. competition, etc, each contestant using the same ammunition).
The object of the exercise is to place the greatest number of pellets onto the paper target, with each contestant shooting from the same position and range using the same ammunition.
At my club the prize is typically a frozen turkey (ready for the oven) for each round of a dozen or so shooters. After the guns are put away we frequently have a few adult beverages, maybe a little lunch. A good time is the goal, and we usually accomplish that goal.
I am both fortunate, and unfortunate, to own a shotgun that repeatedly wins at the turkey shoots, whether I am using it or anyone else uses it. Consequently, there is always at least one round in which everyone gets to use my shotgun. The scoring is lots of fun.