I don't know how others do it, but I use a digital camera - nothing fancy, but one with a relatively good resolution. Mine is 3.2 mega pixels. I don't use flash. I've had some luck with natural aquarium lighting with and w/o room lighting. Flash is too intense for a close up. A real cool effect is shooting in a totally darkened room with just the tank light.
I switch my camera to a close-up mode. I aiming the lens to a large object like a rock in the aquarium to pre-focus and lock that focus to a certain distance where I anticipate the fish will be. Then I frame my scene for a nice background and wait for my fish to swim by. I have to be very patient and be ready at the trigger because, as you say, the fish are always in motion. If I don't care about the background, I follow the fish. I found that background makes a difference and try to stay with a contrasting smooth color to make the fish stand out. The important thing to remember, always, is to maintain the distance you pre-focused to the fish, and try to synchronize your camera movement with the fish motion to "steady" the shot.
The beauty of the digital camera is, that I don't waist film by snapping away many frames. However, I don't have a fancy digital camera, so after I get a good shot, I have to do some editing to adjust color and crop it for the correct framing.
Hope that helps.