Basics: The basic process is to take a half full can of beer with the top cut off. Place it where you are going to be doing the cooking and then place the chicken over it so that the beer can is inside the chicken. Since it's advent, people have begun adding all kinds of spices and extras to the beer to make it more flavorful. Generally, ingredients like garlic, cayenne, diced onions, or cumin added but almost any spice or herb can be used.
Why Beer?: So why does this work so well? First of all, you are adding a source of moisture to the chicken that keeps it from drying out. Second, you are adding beer. Now, more than the fact that beer is good, the yeast and malt found in beer reacts with the chicken, particularly the skin, making it thin and crispy while the meat remains juicy. For information on what kind of beer to use check out my friends Bryce's great article on The Beer in Beer Can Chicken. But you don't have to use beer. You can make this with wine or Garland Rome has suggested using crab boil, which works really well.
Grilling vs Smoking: Like I said, you can do this regardless of how you plan to cook the chicken. Of course, it will turn out differently if you put it on the grill versus in the smoker, but the basic principles still apply. Cook the chicken as you would normally. Always put something under the chicken like foil or a baking dish to catch the drippings.
Accessories: The biggest problem you might run into is the chicken tipping over. The weight of a half full can of beer just isn't enough to keep the chicken standing, especially if you are doing a large bird. But never fear, your fellow outdoor cooks have recognized the need and a host of products have appeared on the market to help you out. If you look in the "Elsewhere on the Web" section you will find a few variations to choose from.
Follow the step by step and try: "Making Beer Can Chicken".