Electric Motors
A an electric motor converts mechanical energy into rotational mechanical movement. Lots of this information is from Maxon1)
Brushed DC Motors
Conventional brushed motors used a coil wound around an iron core rotor with leads going to a commutator. External permanent magnets are magnetized in opposite directions and provide a fixed magnetic field.
Maxon makes fancy core-less brushed DC motors that have a thin cylindrical winding that sleeves over a permanent magnet.
Lorentz Force
The basics of electric motor operation relies on the Lorentz force which states that a current conductor in an external magnetic field experiences a force.
Formally, a particle of charge $q$ with a velocity $\textbf{v}$ in an electric field $\textbf{E}$ and a magnetic field $\textbf{B}$ experiences a force $\textbf{F}$:
$\textbf{F} = q\textbf{E}+q\textbf{v}\times\textbf{B}$
Permanent Magnet
Permanent magnet power density has been improved over the years. Current fancy magnet is the Neodymium iron boron.
Brushed Commutation
A DC motor has multiple winding that need to be commutated (switched) mechanically. Two types of brushing options, graphite and precious metals. There's life problem with brushes because of brushfire, which is a rapid event were the brush short circuits two windings for an instantaneous period of time and causes a spark.
Brushless DC Motors
Brushless motors are electrically commutated motors. They have a winding on the inside with a permanent magnetic core.