DC Brushed Motors

The original DC motor and still the best suited for many applications.  

You get bags or torque by giving it current (they develop a maximum torque when stationary, linearly decreasing as speed increases).

Brushed DC motors generally have lots of commutators.  This means they are often superior for low RPM applications compared to brushless DC motors, as brushless can give you a steppy feel at low speed.

Driving of brushed DC motors is nice and simple and in applications needing a good feedback loop it can be possible to do this in simple electronics rather than needing to use a software loop of some kind.

Larger brushed DC motors give you loads or torque at low speed and in some applications are used without gearboxes to remove gearbox friction etc and provide very fast motor response (e.g. haptic feedback, gyro stabilised heads, etc).  Smaller motors need to operate at a faster RPM with a a gearbox to give the same torque output. Planetary gearboxes usually give the lowest friction.  The higher the gearbox ration the smaller the motor can be but the more gearbox friction etc there is.

Downsides of brushless:

Brush wear out

Brushless DC Motors (BLDC or BL Motor)

Also known as electronically commutated motors (ECM or EC motor).

Brush-less motors don't have brushes to wear out and are often superior to brushed DC motros in higher RPM applications. Can give a "steppy' feel at low RPM due to the drive method needed.

Harmonic Drive servo motors

Very good high end servo motors, very expensive!