What type of electric motor drive can be used for keeping a load at specific heights?
This is typically done with an electric brake attached to the motor.  The brake mechanically keeps the motor from turning when not energized.  When you press the button and the motor starts to turn the brake is released so the motor can turn.

If you want finer control over the speed of the motor you can use a Variable Frequency Drive (VFD). This would still use the same motor brake arraingment. VFDs in lifting and hoisting applications almost always have a braking resistor to counteract the ability for the inertia of the load to overdrive the motor.