what are action potentials and how do they work?

Action poteWhen a neuron is not sending a signal, it is "at rest." When a neuron is at rest, the inside of the neuron is negative relative to the outside. Although the concentrations of the different ions attempt to balance out on both sides of the membrane, they cannot because the cell membrane allows only some ions to pass through channels (ion channels). At rest, potassium ions (K+) can cross through the membrane easily. Also at rest, chloride ions (Cl-)and sodium ions (Na+) have a more difficult time crossing. The negatively charged protein molecules (A-) inside the neuron cannot cross the membrane. In addition to these selective ion channels, there is a pump that uses energy to move three sodium ions out of the neuron for every two potassium ions it puts in. Finally, when all these forces balance out, and the difference in the voltage between the inside and outside of the neuron is measured, you have the resting potential. The resting membrane potential of a neuron is about -70 mV (mV=millivolt) - this means that the inside of the neuron is 70 mV less than the outside. At rest, there are relatively more sodium ions outside the neuron and more potassium ions inside that neuron.

 The resting potential tells about what happens when a neuron is at rest. An action potential occurs when a neuron sends information down an axon, away from the cell body. Neuroscientists use other words, such as a "spike" or an "impulse" for the action potential. The action potential is an explosion of electrical activity that is created by a depolarizing current. This means that some event (a stimulus) causes the resting potential to move toward 0 mV. When the depolarization reaches about -55 mV a neuron will fire an action potential. This is the threshold. If the neuron does not reach this critical threshold level, then no action potential will fire. Also, when the threshold level is reached, an action potential of a fixed sized will always fire...for any given neuron, the size of the action potential is always the same. There are no big or small action potentials in one nerve cell - all action potentials are the same size. Therefore, the neuron either does not reach the threshold or a full action potential is fired - this is the "ALL OR NONE" principle.
 Action potentials are caused by an exchange of ions across the neuron membrane. A stimulus first causes sodium channels to open. Because there are many more sodium ions on the outside, and the inside of the neuron is negative relative to the outside, sodium ions rush into the neuron. Remember, sodium has a positive charge, so the neuron becomes more positive and becomes depolarized. It takes longer for potassium channels to open. When they do open, potassium rushes out of the cell, reversing the depolarization. Also at about this time, sodium channels start to close. This causes the action potential to go back toward -70 mV (a repolarization). The action potential actually goes past -70 mV (a hyperpolarization) because the potassium channels stay open a bit too long. Gradually, the ion concentrations go back to resting levels and the cell returns to -70 mV.

The answer above is good, but I like the 'wave' analogy.  At a stadium, everyone is sitting down and whatever.  That is like the resting potential (no stimulus has been applied). The nerve (or crowd) is said to be polarized, which means negative to the surrounding area (or in the seated position).

Then, for whatever reason a stimulus triggers an action potential (it could be a neurotransmitter in the brain, light triggering the rods and cones in the eye, etc).  That would be like the announcer guy saying over the PA, "Hey guys, start the wave in Section A."

Once that occurs, the nerve becomes DEpolarized in one section only (everybody in section A stands up).  This is because of the Na ions that go rushing out and make that part of the neuron positive instead of negative (a person goes from seated to standing position).  

This depolarization in one part of the nerve triggers the next part of the nerve to become depolarized next.(hey, why is everybody in the next section standing up and yelling "whoo?").  So Na ions rush out of the next part of the nerve cell based on charges and all that.  But the original section now has to go back to normal.  It's not the wave if you don't sit back down.  So the first part of the nerve becomes REpolarized (K ions come rushing in and make it negative again).  While section B is standing up, section A is sitting back down.  And so on, each section triggers the next, and you get a wave like action.  Just like random people sitting then standing is not the wave, random parts of the nerve changing polarity is not a nerve impulse.  It takes a sequential, orderly progression of polarization-depolarization-repolarization (sitting-standing-sitting) to make an action potential (wave). You wouldn't see the crowd doing the wave unless people's movement was organized, just like how a cell transmits an impulse.  

The one thing I haven't included is the refractory period.  You'll notice that the nerve starts out negative due to a build up of Na ions on the inside, then locally becomes positive when Na rushes out.  It becomes repolarized not because Na rushes in, but instead K (which is also a positive ion).   The refractory period is the time in which the two swap places again, after the action potential has passed through (after people sit down after the wave they pick up their coats and programs and settle back in). This is why an action  potential will not move backwards along a nerve.