There were many different types of buildings in an ancient Greek city some were private and some were public, some were religious and some were political. The most impressive buildings, and the ones whose ruins are best preserved today are the public buildings.
The most common building in any Greek city was the house.
The most richly decorated room in the house was usually a room called the andron, which means men's room.
Often the largest and most expensive offering was a statue of the god. The statue of Athena in the Parthenon, for example, was made of gold and ivory, was almost 12 meters high and contained approximately 1,144 kg of gold. The statue of Zeus in the temple at Olympia was considered one of the wonders of the ancient world.
Many Greek cities also had a theatre. Greek theatres were always built into the side of a hill, and some were steeper than others. They were usually semi-circular and could seat up to 21,000 people.
Our architecture today is way more advanced than the Greeks' ever was.