get yourself a digital multi meter they aren't too expensive and will help you solve sooooo many electrical problems. when the quad is off it should be reading about 12V. when the quad is running it should be reading about 14v. this is because electricity produced in the magniteo is converted to dc in the regulator rectifier and sent to the battery to charge it.
so what i would do is jump it with the car battery, put the multi meter to the battry cables (with no battery in it if its not working) and you should be getting a charge. if you dont, follow up the wires from the magneito to the regulator rectifier (AC) then the output from that which should be about 14v (DC). you should be able to pin point whats gone bad.

 

I agree with irishdude completely...

I would add only two minor points. The battery voltage while charging should be 14 volts (as irishdude correctly stated) but with a tolerance of +/- 0.5 volts. So that would make the range of acceptable voltages 13.5 to 14.5 volts DC.

The charging system on a 150cc quad is a little wimpy. So do this test at a medium fast speed (like 3000 RPM) and make sure the headlights are off. At idle the stator output is barely able to keep up with the brake light being on. At speed it is barely able to keep up with the headlights on.

 

what setting should the meter be set to?

 

for the voltage on the battery and after the regulator rectifier you set it to 20V (DC) and im not quite sure about the AC coming from the magnito but set it to AC with the highest setting and keep reducing it so you dont blow the meter