Cats did not come from lions. According to history they were cult animals in ancient Egypt, and come in different species hence different sizes, types and colors. Check out http://en.wikipedia.org/wiki/Cat for more information.
Yes, today's domestic cats developed from larger wild cats or "Felidae". Not necessarily lions and other "big cats", but their smaller cousins. All cats, wild & domestic, are very much related, genetically. But each comes from various, very diverse branches of the Felidae family.