You should easily understand how pumps work: when you extend the plunger, water can only come in through one of the check valves, so that's what it does; then, when you push in the plunger, the water must exit through the other check valve and into the PC, where it pressurizes the air. It doesn't get much simpler than that, but it can if you request a better explanation and/or a diagram.
Once you understand the basic pump, you could take a look at some other designs that I have recently proposed. The first, which is a "
Two-Stroke Pump," is quite simple; it's also what I'm going to use in my PCgH. Basically, you have two standard pumps back-to-back, so you fill one chamber and empty the other with each stroke. The diagrams should help you comprehend that.
The other idea, the "
Dual (or Two)-Stage Pump" (don't confuse "stage" and "stroke" in both methods) is relatively more complex, and it's what joannaardway will use in her PCgH. There are a few ways to implement it (my first diagram is
incorrect), and joannaardway's seems to be the best. No matter what method is used, it involves two (or more) pump tubes, each with its own plunger; in one stroke, both or only the widest pump is used, and in the other stroke, only the smallest pump is used (the wider one could be used in the latter stroke if both are used in the former, in order to have a pump volume difference). Basically, you use the widest pump in the beginning (when there's less pressure/resistance) for maximum pressurization speed, and the smallest one in the end (when there's more pressure/resistance) for maximum ease of use and ability to get high pressure. That thread explains this "phenomenon" better than I do here.
Earlier, I support dual-stage pumps, and joannaardway did not (she claimed that two individual pumps are better); now, however, the roles are reversed. For me, using a pump with two parts seems more complicated (especially with joannaardway's solution of one tube inside another), and if you can get individual pumps' input and output check valves to go around each other, that would be the best. For now, I'm content with a simple two-stroke pump (though you could have a two-stroke and dual-stage pump, which would be extravagant). However, I will wait and see what joannaardway has to say about her PCgH's pump, which I agree will enable greater pressures.