Stepper motors are different from ordinary DC motors in at least four important ways.
The first difference you notice is that they have no brushes or commutator (the parts of a DC motor that reverse the electrical current and keep the rotor—the rotating part of a motor—constantly turning in the same direction). In other words, stepper motors are examples of what we call brushless motors. (You'll also find brushless motors in many electric vehicles, hidden away in the wheel hubs; used in that way, they're called hub motors.)
The second major difference is in what rotates. Remember that in a basic DC motor, there is an outer permanent magnet or magnets that stays static, known as the stator, and an inner coil or coils of wire that rotates inside it, which is the rotor. In a brushless hub-motor, the coils of wire are static in the center and the permanent magnets spin around them on the outside. A stepper motor is different again. This time, the permanent magnets are on the inside and rotate (making up the rotor), while the coils are on the outside and stay static (making up the stator).
The third big difference between an ordinary DC motor and a stepper motor is in the design of the stator and the rotor. Instead of one large magnet on the outside (the stator) and one large coil rotating inside it (the rotor), a stepper motor has an inner magnet effectively divided up into many separate sections, which look like teeth on a gear wheel. The outer coils have corresponding teeth that provide magnetic impulses, attracting, repelling, and making the teeth of the inner wheel rotate by small steps. This will become clear in a moment when we look at some pictures.
The final difference is that a stepper motor can stay still, in a certain position, once it's rotated through a particular angle. That's obviously crucially important if you want a motor to power something like a robot arm, which might have to rotate a certain amount and then remain in precisely that spot while another part of the robot does something else. This feature is sometimes called holding torque (torque is the rotary force something has, so "holding torque" simply means a stepping motor's ability to stay still).