Often you want to select a motor controller that can handle the max stall current but this is not always necessary under the right circumstances. If you plan to use a motor controller rated at a lower than stall current it is highly recommended to have some sort of current limiting or protection to prevent damaging the controller (I have seen what can happen without this protection ... it isn't pretty). If you are absolutely certain your load will not never stall your motor then it is possible to operate without this protection, but this is not usually a safe assumption for robotics.
Motor controllers are often given two current ratings one for continuous operation and one for transient operation. The transient rating should exceed your stall current, thermal protection of the controller can help prevent a continuous stall condition from damaging the controller. The required continuous operating current depends on your application and expected loads, motors often have a current-torque curve or constant that describes this relationship.
Some motors will also give a max operating current that is less than the stall current, (this is the max continuous operating current of the motor to prevent overheating) it may be desirable to have a current limiting controller to prevent damage to the motor.
As for the maximum power current, I suspect this is describing a point in the motors operating range that provides the highest power output. This is useful for optimizing the design of a system as it may be desirable to operate at this point. However they could be describing the max continuous operating specs ... without a real datasheet it is not always clear what they are intending to convey ... even with a data sheet it is not always clear. you could contact the manufacture/distributor for a clarification of this spec.