Explain the course of an action potential across a neuron's membrane

The action potential is the change in electrical voltage across a neuron's cell surface membrane, as an impulse is passed along the neuron. This is the way signals are transmitted in the nervous system. It is a result of the movement of ions (mainly sodium and potassium) across the membrane through voltage-gated channels.
In the classical model, the membrane of a neuron begins at its resting potential ~-65mV. This is a result of outward potassium movement through channels. A raise in the membrane potential past the threshold (-50mV) results in positive feedback of sodium influx due to the opening of voltage-gated sodium channels. These same channels are time delayed. When they close, voltage-gated potassium channels also open, resulting in a return to resting potential. This entire process occurs over the course of a few milliseconds.
A common misconception is that the sodium-potassium pump maintains the resting potential. Rather, it maintains the concentration gradient required for action potentials to work.