It depends on the era.

4th-5th century B.C. the preferred armor was a bronze cuirass and helmet like this.

Later on they moved to what is called a "linothorax" for the chest armor. This type of armor would be used throughout the Hellenistic era, eventually giving way to chain and scale armor towards the end. We don't know exactly how it was made, but it was primarily linen bound together with various resins/glues/fats to make it stiff. Helmets/greaves would still be made from bronze. At times some bronze scales/sheets were added around the midsection for more protection/ornamentation.

There was a pretty famous study of its qualities done by the University of Wisconsin. It proved quite effective against bows/swords/spears and was incredibly light. The main downside compared to quilted armor is it was incredibly stiff. Eventually it breaks in a bit to fit the wearer, but it your upper body would be relatively constrained within it. Here's a fun picture of one of the students with an arrow sticking out of him. It was an incredibly good armor for the greek style of combat and the hot climates in which it was used. The hoplites needed something that could ward off arrows and glancing blows, but didn't really need heavy/expensive metal armor due to their large solid shield wall that protected most of the body during melee combat.

This page has links to plenty of documentation on how they made the armor and its effectiveness, complete with instructions on how to construct your own.