To separate the gold from the gravel is relatively simple and anyone can learn how to pan in three simple moves: 1) shovel a bit of paydirt in the pan, 2) dive it into the water and 3) move gently, in a round motion. Once in a while you should change the motion into a front to back long wave, by keeping the pan on a moderate angle, to allow the surficial gravel to escape. Your gold will sink to the bottom and settle there. By repeating these movements you will slowly eliminate all the lighter materials until only gold and some "black sand" (heavy minerals, mostly iron related) will be left on the bottom of your pan.
In a much bigger scale the separation of the gold occurs in the same way: the large washing plants are using the same principle of your pan (gravity, motion and water). The lighter gravel has to be washed away from the gold. For a big scale operations your shovel will be obviously substituted by excavators and dozers: the more is the gravel that you can wash, the more gold will be recovered. Quite simple eh?