Types of stepper motors
The motor I've illustrated here has a rotor magnet divided into lots of alternating north and south poles, so this design is known as a multi-pole motor. The more poles, the shorter the distance the motor rotates on each step and the more precisely it can be controlled.
In a motor, a "phase" usually means one or two opposing electromagnets that operate alternately (out of sync with one another or out of phase, if you prefer). The motor I've illustrated up above has two phases (two pairs of electromagnets, so four electromagnets in total, arranged at 90 degrees). In a three-phase stepper motor, you might have three electromagnets arranged at an angle of 120 degrees (so three individual electromagnets, although a three-phase motor could also have three pairs arranged 60 degrees apart). A four-phase motor has eight electromagnets arranged in four pairs, with each one separated from the next by 45 degrees.
Advantages and disadvantages of stepper motors
The reason for using a stepper motor is to achieve precise control: you can make it move through a defined angle. But there are drawbacks too. Stepper motors can sometimes be quite jerky, because they start and stop each step with a sudden impulse, which isn't always what you want if you're trying to build a precision machine. An alternative to using a stepper motor is to use a servo motor: a motor with a built-in feedback mechanism. Typically, a servo motor has what's called an optical encoder attached to its rotor. In plain English, that's a black-and-white patterned disc that moves in front of something like a photoelectric cell. As the disc turns, the cell detects the black and white pattern and an electronic circuit figures out from this exactly how much the disc has rotated. Using this feedback, the motor can be controlled more smoothly (and typically much more precisely) than a simple stepper motor. Servo motors are much more sophisticated in design than stepper motors and tend to be more expensive, which is why steppers are often used instead.