As you may have guessed a geared motor contains gears that transfer the speed of the motor to the speed of the wheel. I see it similar to the transmission of an automobile but with a much more compact design. There is a single gearcoming out of the motor and that is attached to three plastic planetary gears, larger than the motor gear. This type of motor works at a high rate of speed, having a higher RPM, and they are meant that way. This allows it to provide better acceleration and significant amount of torque. However, the gears tend to wear out faster, so you may consider that fact if you are planning a long term ownership.
The direct drive motor has a simple design just like the geared motor. There is a stator, which remains stationary, and a rotor which rotates around the stator and that is caused by a magnetic field. This is quite similar to operation of an alternator in a vehicle. The stator consists of the axle, that passes through the middle of the motor, and the copper windings. The rotor consists of magnets along the inner circumference of the rotor. When electricity is applied it causes the magnets, therefore the rotor, to rotate around the stator. This rotation results in the entire shell of the motor, and the wheel to turn.