Why did my miter/table saw stop cutting?
Most of these power tools use a type of commutator motor called a universal series motor. Current passes through the windings on the rotor, called an armature, through brushes that run on the commutator which is made up of a number of segments, usually 12-35 or so, arranged in a cylinder. Between each of the segments is one winding that runs in two slots of the armature core, and then there are a number of these windings in series between the brushes at any one time. As the armature spins, commutator segments pass under the brushes and the windings connected to that segment reverse their current direction as they pass under the next field pole to keep the torque they produce all in the same direction.

What can happen is that some of the windings can break, typically near the connection to the commutator, and leave a series of winding sections that are not connected to the other sections. While those sections pass under the brushes, they produce torque, and though it's not a continuous torque, it's enough to keep the motor turning. When the motor stops, it's a matter of chance as to where those breaks in the circuit are relative to the brushes, so it may just jerk into an unenergized position, or do nothing, but manually rotating the armature to a different position may make it start.
   You can usually see when this happens, in that the sparking at the brushes, visible through the vents at the back of the motor, gets significantly worse than normal.

Unfortunately, these winding faults aren't easy to repair, and usually requires a new armature, which then will likely need to be pressed into a bearing, though some service parts come with the bearing already attached, to be fitted into the housing with a retainer plate or C clip.