Can the anti-torque of a motor be used to spin a space vehicle about it's axis? Imagine an electric motor together with its power supply in space. What exactly will happen to it if it is suddenly powered? Will it spin about its axis, or it will experience no motion relative to an observer on Earth?
 A: It will spin about its axis.
In general terms, if you attach a flywheel to a big motor in space and turn it on, the flywheel will spin in one direction with some angular velocity, and the motor and whatever is attached to the motor will spin in the opposite direction with a different angular velocity. This is the basic principle behind a reaction wheel.
Reaction wheels employ the law of conservation of angular momentum, which means that if you start with a net angular momentum of $0kg \centerdot m^2/s$ (nothing is turning) and you add $5kg \centerdot m^2/s$ of angular momentum to the flywheel, the satellite must turn in the opposite direction with an angular momentum of $5kg \centerdot m^2/s$ to maintain a net angular momentum of $0kg \centerdot m^2/s$.
In other words, $L_{wheel}=L_{satellite}$ at all times. If you assume one direction to be positive and the other to be negative, you can restate this relationship as $L_{wheel}+L_{satellite}=0$.
This is sort of like the rotational equivalent of "every action has an equal and opposite reaction".
Reaction wheels usually work very well, but they can become "saturated", which can significantly reduce their effectiveness. In theory, a reaction wheel can function as long as you can add angular momentum to the system. However, in practice, you run into material constraints and equipment limits. For example, your motor may have an upper limit on the number of RpMs it can produce; when getting get close to this limit, the reaction wheel loses accuracy and becomes saturated. To correct this issue, engineers add systems to satellites to bleed momentum from the wheels, slowing them. 
Two popular methods of bleeding momentum are:


*

*Magnetorquers. Devices that apply torque to Earth's magnetic field to keep the satellite steady as the reaction wheels slow. This is a very common way to bleed extra momentum, and is even used on the Hubble Space Telescope.

*A small engine of some sort, to counteract the torque applied when winding down the reaction wheels. This method is often used on satellites where mass is kept to a minimum, and they don't want to carry a magnetorquer; or on satellites where there is not a magnetic field to torque against.
There is at least one more device worth mentioning. The CMG (Control Moment Gyroscope) is used where efficiency is required, or where a reaction wheel cannot provide the required torque. For example, the ISS has a CMG for attitude changes.
A reaction wheel differs from a CMG in one important respect. A reaction wheel changes its angular momentum to effect a rotation about an axis, but a CMG employs the gyroscopic effect to make attitude changes instead. Greatly simplified, a gyroscope resists changes to the axis of rotation. A CMG applies a torque to a large gyroscope, which then resists the torque. When the gyroscope resists the torque, it rotates the station around the gyroscope.
Related or interesting:
Georgia State University's database for physics (conservation of angular momentum.)
Wolfram Research on reaction wheels
Wolfram Research on conservation of angular momentum (they explain it with diagrams and whatnot. It's pretty good.)
Reaction wheels desaturation using magnetorquers and static input allocation.
Reaction wheels (Google Scholar search)
Three-axis attitude control with two reaction wheels and magnetic torquer bars (About regaining the attitude control systems aboard the FUSE satellite in 1999 after the failure of two of its four onboard reaction wheels.)
A: The casing will spin in the opposite direction. That is the principle of reaction wheels.
A: Indeed. It is due to the law of conservation of angular momentum. The angular momentum of the rotating element within the motor will exactly cancel that of the rest of the motor, thereby giving zero net angular momentum, as with the initial conditions.
