Hello there,
I appears, from what I can see of the motor in the very first pic, that your motor is a 110/220 vac Capacitor start, Induction run motor. This type of motor will NOT have brushes. The fact that the motor started, means that the capacitor is/was working, and the windings are good. The reason that it stopped is more difficult.
Did it grind slowly to a stop, like over the course of a minute or so? Or was it more rapid? Was the motor hot to the touch anywhere after it stopped? I am half inclined to say, without a lot of information, that the bearings might be junk, and are simply bound up.
Post a photo of the motor with its nameplate information to give us more information.
Basic electric motor troubleshooting. Eliminate the simple things first.
The first thing to check, is the wiring connections. Make sure that they are all tight, corrosion free, and making contact. Be sure to check the entire circuit, switches, disconnects, fuses, even the plug at the end of the cord.
MAKE SURE you UNPLUG the machine BEFORE checking these connections!! Electricity KILLS!! If you are not comfortable, find a friend or professional who is.
Disconnect the motor from the drive train. Turn the motor on and see if the problem persists. If it does, then we have eliminated the rest of the lathe mechanically, and we can dive deeper into the electrical. If it does NOT, then the problem is mechanical, and you can then begin troubleshooting the mechanics.
Try these things first, and post your results.
-Cody