Corundum is, at its most basic, aluminum oxide. Next to diamonds, corundum is the hardest stone currently known to man. When they form naturally, corundums take the shape of hexagonal crystals that bulge out of their parent rocks. Corundum in its natural state is clear, but when other trace elements seep into it, it changes colors. The stone has been found in a variety of colors from red (which is known as a ruby, and is an extremely valuable stone), to blue, yellow, purple, and others (which are usually referred to simply as sapphires). Only gems with exceptional clarity are turned into jewelry, however.
Corundum is found in both metamorphic rocks and in igneous rocks. Corundum appears in a wide variety of rocks, but it is always found in areas that lack silica, and which are rich in aluminous materials. Igneous rocks which don't have any quartz, and which are deficient in iron and magnesium, are prime candidates for corundum. Corundum is even more commonly found in metamorphic rocks which began life as aluminous or carbonate sediments.
Like most gemstones, corundum is formed fairly deep in the earth. The reason that it's usually found in metamorphic rocks is because of the often intense pressure and heat that those rocks experience to change them. Like diamonds, corundum requires only its base materials (aluminum and oxygen) to come into contact with heat and pressure in large enough amounts. When an igneous rock is made from cooling magma, crystals of corundum may form inside it as the aluminum oxide cools and bonds together with other materials. Likewise, if aluminum oxide is present in a rock that changes and becomes a metamorphic rock, then that heat and pressure may cause the element to change and become crystals of corundum.