By 400 BC, Greek architects had added a third type of column to the old Doric and Ionic styles. People called the new style the Corinthian column, after the city of Corinth. The Greeks never actually used the Corinthian column that much, even in Corinth. But the Romans, who liked fancier buildings, used it a lot.
The Corinthian style is fancier and heavier than the Ionic style. In Corinthian temples, the columns have a fancier base to stand on. At the top of the columns, on the capital, there’s a stone carving of acanthus leaves, under the architrave (ARR-kuh-trayv). On the architrave, as in Ionic temples, there is a continuous frieze where the triglyphs and metopes would be on a Doric temple.
Learn by doing: walk around town and look for different kinds of column capitals
More about Doric Architecture
More about Ionic Architecture
Bibliography and further reading about Greek architectural orders:
Ancient Greek Architects at Work, by J. J. Coulton (1982). An interesting look at how Greek architects worked.
Greek Architecture, by A. W. Lawrence, R. A. Tomlinson (5th edition 1996). Might be a bit out of date.