A nerve cell (neuron) is a cell whose shape and composition enable energy to travel along its surface. This ability is refered to as "action potential", which implies the ability to do actions, such as move ions and manipulate proteins. Note that potential unfortunately is also an electrical term for voltage. The term action potential in microbiology, however, does not refer to the voltage difference between the inside (cytoplasm) and the outside of the cell, although that voltage difference also contributes to the action potential moving across the nerve cell surface.
There are basically 3 types of nerve cells, designated unipolar, bipolar, and multipolar, depending on the number of fibers coming out of the soma, which is the central bulbous part of the nerve cell that contains the nucleus.
Gates. The nerve cell membrane contains protein complexes (gates or channels) that control how sodium and potassium ions, and other molecules, cross into and out of the cell. There are about 27 different types of gates in various proportional concentrations embedded into the nerve cell membrane, but learning about 4 of those types can give a good picture of what is happening.
Each gate has its own way of controlling which ions it allows to pass, and the direction (in or out of the cell) the ions can pass. Each type of gate has its own triggers to control how and when to open/close: usually a given temperature, ion concentration inside/outside the cell, and the average difference in electrical charge (the actual electrical "potential" difference) between the inside and outside of the cell. The movement of ions and other molecules through these gates in turn also affect the temperature and voltage across the gates. So there is constant back/forth interaction and change along the cell membrane as the gates are turned on/off: the temperature/voltage turning the gates on/off, and the gates changing the temperature/voltage, which in turn impact the gates, etc.
Direction. The mechanism that controls the direction of the action potential depends on the gates' inability to react instantaneously to a change in temperature/voltage. For example, once the given temperature or voltage threshold is reached for a gate that prompts that gate to close, it takes time for that gate to be able to open again - even if the temperature and voltage changes to the threshold for opening the gate. So those gates that have functioned are less able to be triggered than the gates that have not yet functioned that way. Therefore, the action travels away from the gates that were just active towards the gates that have not yet been activated. In a nerve cell that consists of long tendrils, the mere fact of being a narrow channel, the fatigue of activated gates moves the pulse down those channels away from the source that caused the initial change.
Propagation. The concentration of each type of ion determines the electrical charge of the area and also the rate how a group of such ions diffuse in all directions away from an area of high concentration. So, as many or fewer sodium or potassium ions transfer from inside/outside the cell, they diffuse away like a cloud that sticks close to the membrane of the cell. The difference in charge between the inside and outside of the cell (across the cell membrane) keeps the ions on the outide of the membrane close enough to the membrane so that when gates open/close, those ions continue to exist in concentrations sufficient to pass into and out of the gates as those gates are triggered to open/close. As the concentrations on both sides of the cell membrane diffuse, those concentrations impact on (change) the temperature and electrical potential difference away from the initial stimulus and thus trigger gates farther away to open/close.
Intensity. The number of gates being triggered and their timing, and the initial concentrations of ions, determine just how long the pulse lasts and how far it travels along the nerve membrane.
Memory. There is probably also a memory element in the cell, which means that the pulse changes the kinds of proteins and protein complexes in the nerve cell and possibly in the membrane itself. That change enables the cell and cell fibers to retain a memory of the kind of pulse it encountered and make it easier or more difficult for the membrane in those areas to transmit pulses in the future.
The cell membrane at the ends of the nerve fibers (outgoing axon and incoming dendrites) contain a concentration of slightly different gates that are structured to open/close based on external triggers, and to allow different molecules and ions to cross into and out of the cell at that location. Rather than depending on temperature and voltage to trigger such gates, it is the presence and concentration of certain proteins that trigger the opening/closing of those gates.
Nerve cells and other tissues adjacent to those ends can produce or accept those proteins and thus pass the energy from the impulse into and out of the cell. For example, when the energy from a pulse reaches the bulb at the end of an axon, that energy causes calcium ions to diffuse into the cell, which in turn causes the release of proteins from the nerve cell terminal into the space between it and the receiving cell (dendrite of another nerve cell, or another type of cell such as a muscle cell). Those proteins start a chemical process that carries the pulse across the narrow space into the adjacent cell. The calcium ions trigger the creation of proteins that, by attaching themselves to the gate of the adjacent cell, open those gates and allow other molecules and ions to move across the adjacent cell membrane, thus transfering the energy into the membrane of the receiving cell.
Speed. How fast can the impulse travel along the cell membrane? The speed at which the impulse can travel varies from 0.1 to 100 m/s. So the speed can be very very fast.
Passing from the conceptual description towards a more accurate representation...