What does it mean to pattern a shotgun?
Patterning a shotgun is a process where you choose a specific load and choke combination and test how the gun shoots at certain distances. Before patterning your shotgun, it's a good idea to familiarize yourself with a few basic terms. 

Basic terms to know:
Shot:
Small pellets contained in a shotgun shell, which can vary in diameter from .08" for #9 shot to as large as .36" for 000 Buckshot. (For comparison, the bullet from a .38 Special is .357" in diameter.)

Choke:

An interior tube at the end of a shotgun barrel that controls shot dispersion (a constriction in the end of the barrel near the muzzle, similar to a funnel). This constriction compresses the shot charge to modify the impact pattern at a given distance.
Chokes typically come in cylinder, improved cylinder, modified, improved modified, and full. The more "open" the choke, the less constriction it has, giving a wider shot pattern. Typical chokes for use on game birds are Modified Choke and Improved Cylinder, with Improved Cylinder being the more open choke.
Note: A cylinder choke produces a very wide shot dispersion, whereas a full choke will provide a much tighter shot pattern. Different chokes are used for skeet, trap, and sporting clays. In hunting, the type of game and conditions will determine choke type.
For more information about chokes see Shotgun Bore Diameters.

Which combination of shot size and choke should I use?
This is where patterning the shotgun comes in. With an unproven shotgun, you should purchase a box of shotgun shells from different manufacturers and with different shot sizes, and take them to a firing range along with any interchangeable choke tubes you may want to try. Some shotguns do not use external choke tubes they have fixed chokes manufactured into the barrel.   Often you will find double barreled (side by side) shotguns have different chokes for each barrel.  An example would be 1/2 choke for the Right barrel and Modified for the Left.  

How do I pattern a shotgun?
Select an area with a safe backstop, and secure a 40" sheet of paper to a target stand. Mark an aiming point in the center of the paper, and set the paper up at the distance you want to test. The distance should be the anticipated distance between you and the your game when taking a shot. Next, shoot at the aiming point from a steady rest.

To analyze the pattern, draw a 30" circle around the densest portion of the pattern. Check the pattern for evenness and uniformity. There should be enough pellets in the circle to make a clean kill, but not so many that the game would be destroyed. 

Experiment with different loads, chokes, and distances to find the right combination for the chosen game.