The main thing the sodium/potassium pump does is to move sodium and potassium ions, Na+ and K+. It technically generates a bit of charge separation in doing this, but you can ignore it for now.
What is important is that because of the operation of this pump, you now have two sides of membrane with different ion concentrations. One side is high in sodium ions (outside the cell), the other side is high in potassium ions (inside the cell).
So far, that doesn't really do anything electrical, but it does take a lot of energy to get this arrangement: the whole time, the pump is moving sodium ions to where sodium ions are high, and moving potassium ions to where potassium ions are high. Moving things against their concentration gradient costs energy. The sodium/potassium pump gets this energy from ATP.
Now, imagine you open some pores in the membrane (we call them leak channels), that are a particular size that only lets potassium ions through. The sodium ions can't move, but the potassium ions can. A few potassium ions move over to the high sodium side, down the concentration gradient for potassium. This doesn't take any energy, it just happens passively. It also doesn't involve very many ions, just a few, so the concentrations don't change much.
However, potassium ions have charge! These are positive charges, flowing out of the cell: that makes the inside more negative.
Also, because they are positively charged, potassium ions would tend to move towards more negatively charged spaces. So, as the inside of the cell gets more negative, potassium won't leave as quickly. There is an equilibrium, where the forces pulling potassium ions out to where potassium concentration is low equals the forces pulling potassium ions in to where the voltage is negative.
The voltage where this equilibrium occurs is called the resting membrane potential, and for many cells is somewhere between -100 and -40 millivolts: that's the voltage generated across the membrane, indirectly, by the sodium/potassium pump.
Purves' Neuroscience is a good general reference textbook for this sort of thing. The Goldman equation is a mathematical way to figure out what the voltage will be if you know the concentrations and permeability.