Let's look at sodium. Sodium is the most common extracellular cation (meaning it has a positive charge) in physiological systems. It has a typical extracellular concentration of about 150 mM and an intracellular concentration of about 15 mM. Thus the extracellular concentration is greater than the intracellular concentration. This means that diffusion wants to push sodium into the cell. Since sodium is an ion, it must not only move in response to the diffusion gradient but also will respond to the membrane potential. If we assume that potassium is in equilibrium (and given the typical concentrations of potassium) that means that the membrane potential of a typical cell is about -92 mV - see previous page. Since the inside of the cell is negative, sodium is pushed inside the cell by the electrical force. Both the diffusion force and the electric force are trying to push sodium into the cell! Can sodium be in equilibrium under these conditions? Since equilibrium means that the two forces (diffusion and electrical) push equally in opposite directions, it should be evident that sodium is not in equilibrium in a typical cell.
If sodium were in equilibrium given its concentration gradient (outside >> inside so that diffusion is pushing it into the cell), the membrane potential would have to exert a force that would push sodium out of the cell. Since sodium is a positive ion, a positive membrane potential would push sodium out. Therefore, if sodium were to be in equilibrium, the membrane potential would have to be positive. In fact, given the concentration gradient of sodium in a typical cell, the membrane potential would have to be +62 mV, this is the equilibrium potential for sodium.
How can it be that sodium is not in equilibrium? The answer is relatively simple, it is not permeable (actually, it is not very permeable). So even though all of the forces are trying to push sodium into the cell, it cannot go in because it is not very permeable. This is one of those pieces of information that you should commit to memory, sodium is usually not very permeable across most cell membranes.
Now, what if we made the sodium very permeable (to simplify this thought experiment, let us also make potassium impermeable at the same time -- thus we can ignore potassium because it cannot move anywhere). Since both the diffusion gradient and the electrical force are pushing sodium inward, sodium will move into the cell (not that it is permeable). Since a positive ion in moving inward, the inside of the cell will be gaining positive charges and thus the inside of the cell is becoming more positive. When will the sodium stop moving inward (DO NOT say "When it reaches equilibrium!")? Well, we know that sodium will no longer move inward when the two forces acting on the sodium (diffusion and electrical forces) are opposite and equal. This will occur when enough sodium enters the cell so that the membrane potential of the cell becomes positive enough to oppose the inward diffusion. When this occurs, the electrical force will now push sodium outward while the diffusion force still pushes it inward - the forces are now pushing in opposite directions. How positive will the cell become? Well, we already know (see above) that with the concentrations across the the cell membrane found in "real" cells, sodium will be in equilibrium when the membrane potential is equal to the equilibrium potential for sodium, which is +62 mV. So by making sodium the more permeable ion, the membrane potential of the cell will switch from -92 mV to +62 mV. Don't forget that the actual amount of sodium that must move to change the membrane potential is very small so we can confidently assume that the concentrations of sodium do not change significantly.