An open differential is one without any kind of limited slip or locking device. The physics of an open differential mean that the torque on both axle shafts is always equal even if the speed is not. Thus, both wheels are always contributing equally to propelling the vehicle with no exceptions. In ideal conditions when both wheels have equal traction an open differential performs well. However, when there is unequal traction available at the two wheels the one with the least traction sets the limit of how much torque can be delivered to both. For example, if one wheel is on ice and can only receive 100 ft-lb before spinning the other wheel, setting on dry pavement, will also only receive 100 ft-lb. In this case, the wheel on ice will spin while the other remains stationary and you're stuck. It can appear that only one wheel is being driven but the key point is that both are at least attempting to move the vehicle by an equal amount since by definition the torque on the axle shafts is always equal.
A limited slip differential uses clutches or fancy gearing to be able to provide 2 or 3 times as much torque to one wheel over the other. In the example above, if the "ice wheel" spins with 100 lb-ft the limited slip differential could send 200-300 ft-lb to the "pavement wheel" which might be enough to move the vehicle. A true locking differential can send infinitely more torque to one axle shaft by essentially no longer being a differential at all when locked.
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