Color in pottery is provided by metal oxides. The simple fact is that these are the only materials that will survive the temperatures pottery is exposed to when it is fired and also have color. Colorants may be in the clay itself, in a slip, in an underglaze layer or in the glaze. Sometimes color is even applied as an overglaze. We'll focus on glaze colorants in this paragraph because that is the way most potters achieve the color effects they want. Even though relatively few metal oxides are used as colorants, the variations in color that are achievable are almost endless. This is because the resulting color of the pot is affected not only by the metal oxide and its concentration, but by such things as the way the oxide interacts with other glaze materials both physically and chemically, the temperature to which it is fired and the environment within the kiln (oxygen rich vs. oxygen poor) during glaze firing. The most common colorant is iron oxide (what we all know as rust). Iron oxide is abundant in nature and very inexpensive. It normally results in colors than range from light tan to dark brown or even black. Copper oxide (which is usually added to a glaze in the form of copper carbonate and changes to copper oxide during firing) gives greens and blues and colors in between, but in reduction firing it can give a copper-red or oxblood color. Cobalt oxide gives strong blues at very low concentrations. Chromium is a versatile coloring agent and under the right conditions can produce red, yellow, pink, green or brown glazes. Examples of other metals that are used as colorants in glazes are manganese (purple or brown), nickel (brown or gray), vanadium (yellow) and titanium (white).
Some potters use mixture of oxides that have been compounded into a material call a "stain". These are added to a base glaze and can give far more predictable colors which are often more uniform over the surface of the pot. The potters who use stains usually do so because they like that predictability. Other potters like the variability and occasional surprises that come from using colorant oxides. I am in the latter camp--I rarely use a stain.
If you are curious about how a potter achieved a certain color, ask. Most are quite willing to discuss their glaze composition in general terms; however some consider the exact composition to be their own trade secret because they have spent many years developing it.