The simplest setup would be a geared DC motor with a cam attached to the output shaft, as well as whatever it drives.
(Gearing reduces the speed and gives things time to happen, plus vastly increasing the motor torque).
The cam needs to operate a microswitch at the "zero" position of the shaft rotation. The switch breaks the circuit and stops the motor.
The trigger switch just connects across the stop switch to make the motor run past the stop position, so the stop switch releases and keeps power on to the motor.
That's how simple mechanical car windscreen wipers stop in the "parked" position - a contact or switch that keeps power on until they get to the right point.
You could also use the stop switch to operate a relay that controls power to the motor. Having that connect a low value resistor across the motor when it releases would give a more accurate fast stop rather than the motor coasting on a bit.
Example geared motors; the type you need depends on how fast it must and how much power the output shaft needs:
N20 DC geared motor, 3V, 6V, 12V various RPM. · UK based supplier.
www.ebay.co.uk
The worm gear motor with self-locking, ie, in the case in the absence of electric motors. 1 Geared Motor. Load torque: 10KG. No-load speed choice:0.6rpm/10rpm/25rpm/100rpm/120rpm. Material:416 Stainless.
www.ebay.co.uk
• Low speed, Can be forward and reverse. • DC Deceleration. Good Good quality.
www.ebay.co.uk
If it's something that must operate fast. eg. less than a second for a turn, you are probably going to need a more sophisticated and much more complex system.