The electric motor is a standard DC electric motor. See "How Electric Motors Work" for a description of how an electric motor works, and "Inside an Electric Motor" to see the parts inside an actual electric motor.
On the end of the motor is a small 8-tooth gear. This gear fits into the center of the planetary gear system, as shown here:
This gear system is the heart of any electric screwdriver. An electric motor by itself is a pretty weak device. You can grab the axle and stop a small motor's rotation very easily. But if you gear the motor way down, it can have enough strength to drive a screw into a piece of wood effortlessly. In this screwdriver the dual planetary gear system has a 68:1 reduction ratio. With this reduction ratio, the motor will turn 68 times for the chuck to turn once. This means that the chuck moves very slowly relative to the motor, but that the chuck has a great deal of torque (it takes 68 times more strength to stop the motor from spinning because of the gear ratio). The gear system is actually in two layers, and you can barely see one of the gears in the second layer in the following picture:
The top layer of the gear system turns once for every 8.25 revolutions of the motor. The second layer is identical and turns once for every 8.25 turns of the top layer. That gives a total gear ratio of about 68:1.
For more information on how electric motors work, please click here.
For more information on how gears work, please click here.