To expand on what Bass said, the most common means of separating metals from waste material is gravity separation. Prospectors who are out to find gold, for example, will first classify the material using screens in order to try to get the particles the same size. Gold is 19 times heavier than water and also much heavier than most of the waste material. When the particles are the same size and subjected to aeration in water, the heavier particles (gold) will sink to the bottom of the container, allowing the lighter material to wash out. The most common example of this is called "panning". Panning is one of the oldest and still the most reliable means of separating metals from waste material. There are several tools prospectors use for gravity separation. The pan, sluice, panning wheel, "blue bowl", "gold cube" and shaker table. Googling these items should help you to visualize and help understand them. With the modern day gold rush that is being driven by the high gold prices currently, there are several new designs in gold recovery products being developed, tested and implemented in the field. In all processes, the final step to clean up is still the good old gold pan. Adaquate classification and aeration is the key to success.
The floatation process is also very effective but requires all the material to be crushed (pulverized) and chemicals are added to cause the precious metals to float where it can be collected and separated from the waste material. This type of process is usally done by large mining companies with a large facility.
In the old days, mercury was used to recover very fine gold. Gold has an affinity towards mercury (it is attacted to it). Prospectors would use mercury to collect gold particles and then retort the mercury (heat it until mercury vaporizes) and the gold would be left in the container. This is a VERY dangerous process today, and using mercury to capture gold is illegal. Google gold amalgamation.
Diamonds are a bit different. They are heavy too, but not as heavy as gold. They are not as easy to separate using the gravity separation process. For diamonds, grease is implemented. Diamonds have an affinity towards grease and will stick to it. I'm not sure if this will also work for corundum minerals.