The Box Stance - The box stance is the most widely used stance in umpiring. Square up to the plate, one foot slightly forward (towards the batter's back foot), knees flexed, back slightly bent, and completely balanced. This stance can be used over the center of the plate or in the slot (to find the slot position your body so that your nose is centered over the inside edge of the plate). As the pitcher comes forward to deliver the baseball drop smoothly in a crouch. Most umpires crouch to a level that their eyes are at the top of the strike zone. More experienced umpires who use this stance like to stay higher in the zone to prevent being obstructed by the catcher. Below are two great examples of MLB umps using the box.
The Knee Stance - Occasionally you will find umpires who "work the slot" will kneel down on one knee. The umpire will place his knee on the ground behind the catcher. Tall umpires favor this stance. This stance has the advantage of relieving back pressure and allows the umpire to get low in the zone (especially good for 9 year olds). Disadvantages include more pressure on the legs and decreased mobility.
The Scissors Stance - This stance is more common in professional ranks and less common in the amateurs. Instead of kneeling the umpire extends one leg behind him. Some umpires report neck strain from this stance and it is a difficult stance to master.
The Slot Stance - The slot stance is taught by the Jim Evans Academy of Professional Umpiring. It is basically the box stance except the front foot is one shoe length in front of the foot behind the catcher. The feet are also placed wider apart and is more comfortable. The slot is a very comfortable stance for those umpires that "work the slot." Below is an amateur umpire working an exaggerated slot stance. This exaggerated slot stance is used by many fastpitch softball umpires.
Here is a more conservative slot stance: