Did humans evolve from monkeys or apes?

It's a popular misconception that Darwin's evolutionary theories state that humans evolved or descended from apes. That just isn't true. What his theories really state is that humans, apes, and other hominoids (gorillas, chimpanzees, and so on) all evolved from a common ancestor.
Think of your family tree. You and your aunt descend from a common ancestor — your grandparents are her parents — but you don't descend from her. In evolutionary terms, you represent Homo sapiens (humans), and your aunt represents, say, Pan troglodytes (chimpanzees). The common ancestor of humans and chimps isn't as close in time as you are to your grandparents, of course — evolution can be an extremely slow process — but Darwin's theories state that, if you go back far enough in time on that evolutionary family tree, the branch that leads to humans and the branch the leads to chimps converge.