A traditional motor has a series of coils which are automatically switched on and off by a set of brushes in contact with the commutator (where the sparks and ozone come from). Once power is applied, the motor runs itself at a speed proportional to the voltage and the load.
A stepper motor has no commutator. Instead, there are five or six wires coming out of the motor; one wire for each coil (usually four) and one or two common ground wires. Power must be applied to one coil after another in the proper sequence in order to get the motor to turn. In order to obtain the maximum torque, two coils are always on at any time.
Each step only turns the shaft a degree or two. This four step cycle has to be repeated about 50 times for a full revolution (not just once, as shown in the diagrams). If all four coils are switched off, the motor will be free to idle. Otherwise it is always locked in its current position. If the load on a stepper motor is too great or if the stepping sequences are being cycled too fast, it will skip a step.
There are other types of stepping sequences that provide smoother motion or lower power consumption.
One potential problem is determining which wire is which on the stepper motor. The simplest method is to use an ohm meter to figure out which wire or wires are the common ones. Then plug the remaining ones into the controller randomly. There are only 24 possible permutations and 8 of them are correct, so it won't take too long to find one that works.