Why do electric motors draw current even when they are not moving? After testing electric motors, I realized that these motors draw current even when they are not moving. I do feel that these motors are trying to move, but they are not really moving, in the end. 
Why do these motors draw current even though they are not moving? 
 A: Basically motors without rotation act as a simple Resistor between the power source, so it is always a closed circuit and current flows.  
In motors, when forced not to rotate, this is called stalling.  Only some special motors can sustain long periods of stalling, called Torque Motors, and are used to apply a force while speed is zero or very near zero.  
There are many types of motors, but when stalled, you are basically forcing them the biggest current, as they are in an overload condition.
A: In fact, typically a motor will draw much more current when stalled than when running. This is because when running, it is also acting as a generator, creating an EMF which opposes the applied EMF and reduces the overall current. As more mechanical load is applied, the motor slows, the back emf decreases, and more current is drawn. If sufficient mechanical load is applied, the motor stalls/stops rotating, and therefore there is no back EMF at all, and the current is limited by the supply or by the resistance of the windings.
Incidentally, a similar phenomenon occurs in loudspeakers. The input signal causes the cone to vibrate, but the voicecoil moving in the magnetic field also generates a voltage which opposes the incoming signal. When the incoming signal stops, the diaphragm would tend to continue to vibrate, but the loading effect of the amplifier output on the current generated by the speaker damps this effect, improving the definition of the sound. This is one of the reasons why speaker cables need to be so low resistance, since resistance in this path reduces the damping effect.
A: To understand this, you first need to understand how an electric motor works.

An electric motor is, grossly oversimplified, a bunch of electric coils and magnets. There are different kinds of electric motors with different arrangements of coils and magnets. Sometimes the magnets are permanent-magnets, sometimes they are electromagnets, also made from coils. But in the end, the parts which draw current are the coils.
A coil is an electric wire wrapped in a helix. When you put a voltage on the ends of a wire, the wire draws a current. When there is a current in a wire, a magnetic field is created around it. By wrapping a wire in a spiral, the magnetic fields of the individual windings add up and create a linear magnetic field along the coil. Interacting magnetic fields create motion. And that's how a motor moves.
When the motor can not move for some reason, like when it is obstructed somehow, this is no reason why the wires which make up the coils should suddenly increase their resistance. They still allow current to go through them. You might wonder where all this electric energy is going when there is no kinetic energy being created. It gets converted into heat. The coils will heat up. Some motors are designed to withstand that heat buildup, but others are not. They will overheat, get damaged and might in the worst case even catch fire. So don't obstruct an electric motor, unless you know it's designed for that.
A: To simplify:  when a motor is not moving, electrically it looks like a short circuit.  The only limit to the motor current is the resistance of the wires and motor brushes (if it has brushes).  This is the "stall current".  Typically a stall current is very high and will quickly overheat and damage the motor.
When a motor is moving in the intended direction, it acts like a generator or alternator and creates "back EMF".  This essentially reduces the voltage across the motor and reduces the current flow.
Some types of motors (stepper, brushless DC, etc.) are different.
