The Ancient Age is the time from the last centuries of Pre-history, usually 4000 BC in Civilization games, and lasting to just after the fall of the western Roman Empire in about 500 AD. The year 500 AD began a thousand year period called the Middle Ages.
The Ancient Age officially began when writing was discovered. But the game begins a few centuries before this. This is done to give you a settler that's wandering and you can have this "I'm a wandering nomad" feeling in the game. Most parts of the world in real life had remained in a state of Pre-history even by 2000 BC so it is true that not all parts of the world were taking part initially when the ancient age had begun. The first city to be built in real life was Ur (Circa 3400 BC) in the ancient kingdom of Sumer in the region of Mesopotamia, what is now Iraq. The next notable event was in 3100 BC when Egypt was unified under king Menes.
All civilizations in Civilization III start in the ancient era if not customized. Many of the empires that started in this era were around Europe, North Africa, and the Middle East.
Ancient Times are followed by the Middle Ages in Civilization III.