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.