Obviously, you trade off aperture vs shutter to maintain the correct exposure. There are certain specific circumstances where you want to choose shutter speed: blurring waterfalls 1/8 or less, not blurring bird wings, or freezing athletes in motion where you need 1/1000 or more. There are other specific circumstances where you want maximum or minimum Depth of Field, as long as your shutter speed is in the proper range, let's say faster than 1/125 to avoid blurring people who are walking.
The third knob is ISO. On film we wouldn't dial ISO around on every shot, but on a fully automatic, digital camera, you set aperture or shutter and the camera will optimize the other two parameters via some algorithm.
That gets confusing; well, for me it does, as I'm not very good at keeping track of three parameters. I admit that my photography would be much improved if I spent a year with a manual, film camera, (maybe a Leica as suggested by Mike Johnston at TOP) in order to gain better understanding and control. From Juan V's comments I get the feeling he has done exactly that.