The article How Electric Motors Work explains how brushed motors work. In a typical DC motor, there are permanent magnets on the outside and a spinning armature on the inside. The permanent magnets are stationary, so they are called the stator. The armature rotates, so it is called the rotor.

The armature contains an electromagnet. When you run electricity into this electromagnet, it creates a magnetic field in the armature that attracts and repels the magnets in the stator. So the armature spins through 180 degrees. To keep it spinning, you have to change the poles of the electromagnet. The brushes handle this change in polarity. They make contact with two spinning electrodes attached to the armature and flip the magnetic polarity of the electromagnet as it spins.
This setup works and is simple and cheap to manufacture, but it has a lot of problems:
The brushes eventually wear out.
Because the brushes are making/breaking connections, you get sparking and electrical noise.
The brushes limit the maximum speed of the motor.
Having the electromagnet in the center of the motor makes it harder to cool.
The use of brushes puts a limit on how many poles the armature can have.
With the advent of cheap computers and power transistors, it became possible to "turn the motor inside out" and eliminate the brushes. In a brushless DC motor (BLDC), you put the permanent magnets on the rotor and you move the electromagnets to the stator. Then you use a computer (connected to high-power transistors) to charge up the electromagnets as the shaft turns. This system has all sorts of advantages:Because a computer controls the motor instead of mechanical brushes, it's more precise. The computer can also factor the speed of the motor into the equation. This makes brushless motors more efficient.

There is no sparking and much less electrical noise.
There are no brushes to wear out.
With the electromagnets on the stator, they are very easy to cool.
You can have a lot of electromagnets on the stator for more precise control.
The only disadvantage of a brushless motor is its higher initial cost, but you can often recover that cost through the greater efficiency over the life of the motor.