I need to create a tensioning system for a prosthetic device.

Is Arduino something that could power a gear or wheel with a cable or cord attached to it. It would need to be small, deliver torque and have slow rpm's. Thank you for your time in advance.

It could command a servo with the needed performance easily enough.

Is Arduino something that could power a gear or wheel with a cable or cord attached to it. It would need to be small, deliver torque and have slow rpm's.

Note that you used the word "power" and Crossroads answered using the word "command" because the arduino is uController , not a power supply and as he stated, it is capable of controlling motor/servo systems that are powered from an alternate power source.
(such as a battery, possibly Lipo but not necessarily)

So, Arduino is a programmable circuit board that could control a motor that was powered by a battery, correct? Would it be able to communicate with a software app or remote control of some sort? I am looking to create a way to control a small motor in a device that is worn under clothing. Once again, thanks in advance for your time.

No, to control a motor you need an h-bridge to drive current in 2 directions in the motor.
Arduino can control the h-bridge.
Servo is another kind of motor with the controller built in.
Yes, arduino has communication capability, to another device via wires, or via RF with the addition of a bluetooth module for example.

teamLIM:
I am looking to create a way to control a small motor in a device that is worn under clothing.

Start by picking the motor/servo that you want to control. A servo will be much easier to control than a DC motor since it contains all the related driver electronics and gives you direct positional control (or direct speed control, with modification).

Thank you for your insight. I will research "servos" and mini motors and return for advice once i decide which device i will be using. This is exciting.

You might need a linear actuator.
If you want a tensioner then you would need a geared motor assembly that can generate some torque.
A mechanical drawing might help.