Around 1894 BCE (Before Common Era), a small Amorite kingdom established itself in Mesopotamia with a minor city, Babylon. This first age of the Babylonian Empire was ruled by the Amorite dynasty until approximately 1595 BCE. For about one hundred years, Babylon remained a minor city until the Amorite ruler, Hammurabi (arguably the most famous Amorite ruler), set about expanding the town into a city fit for kingship. Under Hammurabi's rule, the empire expanded greatly. However, his rule was short lived, lasting only from 1792-1750 BCE. This area of Mesopotamia was quite vulnerable to attack and after his death, the empire started to disintegrate and shrink until ultimately Babylon was sacked by the Hittites in 1595 BCE.
From about 620-539 BCE, Babylon was restored to its former glory under the rule of Nebuchadnezzar II (among others) until it was conquered by the Persians around 539 BCE. This age is called the "Neo-Babylonian Empire" or "Chaldean Era."