What is a membrane potential?

The existence of membrane potential is just that: as long as they are alive, cells try to keep their cytosol different from the outside soup, mainly by expelling sodium ions. There is a longer story, with cells pumping other ions in or out, or leaking ions due to the imbalance, but resting membrane potential really revolves around sodium. There is an imbalance in sodium across the membrane, and that makes sodium naturally try to balance out, by trying to leak back in.

I am sure you have heard about the famous sodium/potassium ATPase. It expels sodium ions, while letting in fewer positive ions. It is just the most studied way the cells are going about making sure their innards poorer in cations, that is, negatively charged when compared to their environment.

All these pumps are ATP-dependent, and you would know ATP is the main energy store. So as long as cells live and make ATP, they can maintain this difference between their innards and the environment. Once they die, ATP is gone, pumps stop, and there is no sodium disequilibrium - no membrane potential.

Diffusion potential is an artificial concept that we use in order to make sense of the membrane. We imagine a membrane that opposes sodium movement alone, and we calculate how hard will it be (how much energy must be spent) to keep the asymmetry of low sodium inside - higher sodium outside, were a hole punched in the semipermeable membrane. Luckily, we can use some physics and maths, and figure that this energy is dependent of the ratio of sodium concentrations ([Na]in divided by [Na]out). For typical cells, there are 150 mM Na outside and 15 mM Na inside a cell; physics tells us it will take 62 milli-electronvolts to hold a Na ion out once some hole is found.

If you don't provide those 62 meV and you don't close the doors for leaking, sodium ions will fly in. And that is indeed what happens in depolarization, when sodium gets a chance to move.

These electronvolts are a unit of energy, or work. We can think of a few ways of performing this work on a Na ion that tries to leak in: heat the cell until ions stop caring about direction of movement, hold the outside Na in some tight chemical structure etc. But the simplest way is to apply an electric field. If you add to the concentration imbalance a 62 mV electrical difference in electrical potential, you stopped the flow of sodium either way. Get a bit less than 62 mV, and the Na ions will carry on their old ways and try to no avail to leak into the cell. Get a bit above 62 mV, and the flow of sodium will be force in the opposite way. Stay on 62 mV and you achieve equilibrium potential: neither direction will have the upper hand.

The picture is way more complex once you factor in the other ions, but the facts are:

• resting and action potentials are voltages one can measure across membranes, quite often during cells' lives
• diffusion potentials are part of an abstract model
• equilibrium potential is something one could impose on a cell - not something that you'd find in natural frequently, yet not plain fiction.

To understand what a membrane potential is, you should understand the units for -80mv. The negative sign is relative to the observer, yourelf, to the inside of a cell. From your perspective, the inside has an excess of negative ions. On the outside of the cell, there is an excess of positive ions.

Unlike electricity, where charge separation is quantified by electrons, action potential is quantified by ions. Ions are matter, like Na+, K+, Cl-. When electricity flows through a wire, electrons are flowing from excess negative charge to a deficiency of negative charge.

When a cell establishes an action potential, it moves Na+ outside the cell, which moves the potential from 0mv, negative. The number -80mv, or whatever potential that may be observed, is the point at which a net flow of positive ions are not leaving the cell.

Na+ is actively pumped outside a cell, using energy. K+ is passively driven into the cell by simple diffusion to neutralize the charge imbalance. Opposing the outward flow of Na+ and inward flow of K+ is their diffusion gradients. Ions flow from a high concentration to low concentration.

At -80mv, the Na+ concentration is high outside the cell. Intracellular Na+ is being pumped out to counteract any leaking back in. The K+ concentration is high inside the cell from its passive diffusion to cancel the electrical potential.

The physics of why K+ does not neutralize the -80mv is in its concentration gradient. K+ is being pulled outside the cell by this gradient and driven inwards by an electrical potential. The net electrical potential across the cell membrane is -80mv for a given cell's capacity to maintain the concentration of its intracellular ions.

Membrane potential AKA the membrane electrochemical potential, may be thought of as the voltage that is experienced across the membrane, analogous to a battery.

The extracellular side is typically positive, while the cytoplasmic side is typically negative. This induces the electrochemical gradient with ions having a preference for a particular side depending on their charge.

An equilibrium of ions can be maintained by transport pumps which "force" a difference in electrochemical potential on either side of the membrane potential. If the membrane is permeable to a certain type of ion, the the diffusion processes occurs as ions diffuse to their "preferred" location dependent on ion concentrations.

Wikipedia has a nice succinct definition of the functional advantages of maintaining a membrane potential:

First, it allows a cell to function as a battery, providing power to operate a variety of "molecular devices" embedded in the membrane. Second, in electrically excitable cells such as neurons and muscle cells, it is used for transmitting signals between different parts of a cell.

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This is an advanced concept, so perhaps a few video tutorials might explain better than my words. Start here perhaps.