When chlorophyll is abundant in leaves, the green color covers up other pigments. The other colors that were present but blocked by the green become visible when there is a lack of chlorophyll.
Yellows and oranges
These pigments are made by xanthophylls and carotenoids (the same as in carrots) and absorb the green light the cholorophyll misses.
Reds and purples
Some leaves turn red from anthocyanins, a chemical manufactured from the sugars trapped in the leaf.
When all the pigment breaks down, only the brown tannins remain.
Sources: U.S. National Arboretum, State University of New York College of Environmental Science and Forestry, McGraw Hill Concise Encyclopedia of Science and Technology, Simply Science, The Plant Doctor, U.S. Forestry Service