I went to the trouble of learning this just now, so I might as well share:
The most basic electric motor is to have two electromagnets which can rotate with respect to each other. You can make one fixed, but the other has to rotate while still being powered. This requires an electrical connection for a spinning part, and the easiest way was "brushes", apparently. These are literally just metal wires with some spring to them so they can maintain some contact with a rotating part. As you can imagine, this leads to frictional losses and wear, so low reliability.
Another option is to replace the spinning electromagnet with a spinning permanent magnet. That way you don't need the electrical connection, but you do need expensive rare-earth metals to form the permanent magnet.
And then there is the option mentioned in the article, where the spinning permanent magnet is not made of expensive rare-earth metals, but rather of a metal in which a magnetic field is induced by stationary windings.
A lot of modern motors are brushless, and instead of using brushes, they use a controller that monitors the position of the rotor and flips the polarity at the appropriate times.
The methods for monitoring the rotor vary, so you can essentially you can think of brushed motors as a special case of brushless motors. In a brushed motor, the "controller" is essentially a physical switch that flips back and forth as the rotor spins. But physical switches wear out, so less wear-prone methods like measuring back-EMF can make it last a lot longer.
Well, what you say is definitely true but is actually tangential to what I was getting at.
In an old-fashioned brushed motor, the brushes serve two engineering purposes: (1) they allow an electrical connection to a moving part and (2) they automatically accomplish the electrical switching necessary to drive the motor through a full cycle. By "automatically", I mean that they do it by virtue of their physical placement and construction, without any need for computer control.
In distinguishing between brushed and brushless motors, I was concentrating on (1). Brushed motors are simpler, but have flexing metal strands which rub, causing friction and wear. In my mind, this is the primary reason to move from brushed to brushless motors. Now, in addition, when you remove brushes (and make the central spinning magnet a permanent magnet rather than electromagnet) you also get a chance to manually control the switching of the power--per (2)--to the fixed, exterior electromagnet. And this has efficiency advantages which you mention.
I'm sure you know all that, I just wanted to explain my reasoning and be pedagogical for others. Please let me know if I messed up.