When a cell is not firing, it is "at rest." The resting potential of a cell is the difference in charge between the inside of the cell and the environment outside it. For the average human neuron, the resting membrane potential is approximately -70 millivolts, meaning the inside of the cell has a charge that is 70 millivolts lower than the outside.
While a resting potential occurs when a neuron is at rest, an action potential occurs when a neuron is firing.
Neurons send signals down their axons, long arms that conduct electrical impulses down the neuron (typically away from the neuron's cell body.) An action potential occurs when neurons send these signals down their axons, causing a short, fast increase in electrical activity. Scientists therefore often refer to action potentials as nerve "spikes" or "impulses."
Action potentials are caused by the passage of ions through a membrane; the electrical current that results from this passage is how neurons send signals.

When a neuron receives a stimulus, a sodium channel opens first. As the inside of the cell is negative (remember the -70 mV resting potential!), positively-charged sodium ions waiting outside the cell move into the cell through this open channel.
As the positively-charged ions enter the cell, the charge inside the cell increases slowly at first. If the potential inside the cell doesn't increase to a certain firing threshold (in human neurons, -55 mV), the result is a small, localized increase in potential called a graded potential. The neuron doesn't fire in response, and the signal is not sent down the neuron's axon.
If, however, the firing threshold is achieved, voltage-gated sodium channels open and the resulting rapid influx of sodium ions causes the spike in electrical activity that we call an action potential. The charge inside the cell rapidly approaches 0 mV, then crosses it to reach a maximum potential of about +40 mV. This rapid increase in the cell's potential is called depolarization.
In response to the increasingly positive charge inside the cell, the voltage-gated sodium channels close and potassium channels slowly begin to open. As the cell is now positively charged relative to the environment outside it, positively-charged potassium ions inside the cell exit through the channels.
The action potential then decreases back down to -70 mV, a process called repolarization, and actually passes a little bit below it (hyperpolarization), as the potassium channels stay open slightly longer than necessary.
Ion levels within the cell gradually balance out to resting levels with the help of Sodium-Potassium ion pumps, and the cell returns to its -70 mV resting potential.
All action potentials for neurons of the same size will always have the same magnitude. So in the case of human neurons, the action potential will always reach +40 mV, no more, no less. So how do our nerves convey signals of different intensities?
While the magnitudes of these action potentials don't vary, their frequencies can. The more action potentials occurring in a given time frame, the more intense a signal the neuron is sending.
For a great visual explanation of these concepts, watch the Crash Course video here.