What are the stall and free currents of an electric motor? For example, this Vex motor lists its stall and free currents at the bottom of the page.

I think I understand the general idea, but a detailed description would be helpful.

Short answer

Stall current is the maximum current drawn1, when the motor is applying its maximum torque, either because it is being prevented from moving entirely or because it can no longer accelerate given the load it is under.

Free current is the current drawn when the motor is rotating freely at maximum speed, under no load2 other than friction and back-emf forces in the motor itself.


1: Under normal conditions, i.e. the motor isn't being asked go from max speed in one direction to max speed in the other.
2: This assumes the motor is not being driven by external forces.
Long answer
Stall Current
From the Wikipedia page on Stall Torque:

Stall torque is the torque which is produced by a device when the output rotational speed is zero. It may also mean the torque load that causes the output rotational speed of a device to become zero - i.e. to cause stalling. Stalling is a condition when the motor stops rotating. This condition occurs when the load torque is greater than the motor shaft torque i.e. break down torque condition. In this condition the motor draws maximum current but the motor do not rotate.The current is called as Stalling current.

...

Electric motors
Electric motors continue to provide torque when stalled. However, electric motors left in a stalled condition are prone to overheating and possible damage since the current flowing is maximum under these conditions.
The maximum torque an electric motor can produce in the long term when stalled without causing damage is called the maximum continuous stall torque.

Thus from the specification of this motor
Stall Torque:  8.6 in-lbs
Stall Current: 2.6 A

we can see that if the motor is required to apply more than 8.6 in-lbs of torque the motor will stop moving (or accelerating if working against friction) and will be drawing the maximum 2.6A of current.
Although it doesn't say what kind of motor it is, I would expect it to be a Brushed DC electric motor given it's two wire interface.

As an unloaded DC motor spins, it generates a backwards-flowing electromotive force that resists the current being applied to the motor. The current through the motor drops as the rotational speed increases, and a free-spinning motor has very little current. It is only when a load is applied to the motor that slows the rotor that the current draw through the motor increases.

From the Counter electromotive force wikipedia page:

In motor control and robotics, the term "Back-EMF" often refers to using the voltage generated by a spinning motor to infer the speed of the motor's rotation.

Note however, as DrFriedParts explains, this is only part of the story. The maximum continuous stall torque may be much lower than the maximum torque and thus current. For instance if you switch from full torque in one direction to full torque in the other. In this case, the current drawn could be double the continuous stall current. Do this often enough, exceeding the duty cycle of the motor and you could burn out your motor.
Free Current
Again, looking at the specification:
Free Speed:     100 rpm
Free Current:   0.18 A

So when running freely, without load, it will rapidly accelerate up to 100 rpm, where it will draw just 180 mA to maintain that speed given friction and back-emf.
Again however, as DrFriedParts explains, this is also only part of the story. If the motor is being driven by an external force (effectively a -ve load), and thus the motor is turned into a generator, the current drawn may be cancelled out by the current generated by the external force.
