Building an electric motor from a coil of wire, some magnets, and some paper clips is a rite of passage for many budding science buffs. These motors are simple brushed motors. That is, the electromagnet spins towards a permanent magnet and the spinning breaks the circuit, allowing the electromagnet to continue spinning from inertia. Eventually, the connection completes again and the cycle starts over. Real brushed motors commutate the DC supply current so that the electromagnet changes polarity midway through the turn. Either way, the basic design is permanent magnets on the outside (the stationary part) and electromagnets on the inside (the rotating part).
Brushless motors flip this inside out. The rotating part (the rotor) has a permanent magnet. The stationary part (the stator) has multiple electromagnets. By controlling the electromagnets, the rotor spins. With no brushes, these motors are often more efficient, they don’t generate as much electrical noise, and there is no danger of brushes wearing out. In addition, the electromagnets staying put make the motor easier to wire and, if needed, easier to cool the electromagnets. The principle of operation is similar to a stepper motor. Steppers are usually optimized for small precise steps. Brushless motors are optimized for spinning, not stepping.
[Axbm] built a clever brushless motor out of little more than PVC pipe, some magnets, wire, and iron rods. The plan is simple: construct a PVC frame, build a rotor out of PVC and magnets, and mount electromagnets on the frame. An Arduino and some FETs drive the coils, although you could drive the motors using any number of methods. You can see the whole thing work in the video below.
One interesting tidbit is the winding of the coils. [Axbm] put 600 turns of wire around each iron core (soft iron is best, although you can use stainless steel, which is easier to find). Instead of cutting the wire when done, he simply moved on to wind the next magnet. To keep the magnetic fields in the proper orientation, the winding of one magnet has to be in the opposite direction of the previous winding. So if you wind one magnet clockwise, the next magnet must be anticlockwise.