About 2400 BC, the Amorites were living in what is now southern Turkey and Syria. Amorites means “westerners” in Sumerian, and that makes sense because the Amorites lived to the west of Sumer. They spoke a Semitic language, and lived partly in cities and partly as nomads.
We hear about the Amorites mainly from their neighbors in Mesopotamia. Mesopotamians complained that the Amorites were always attacking them. The Akkadian king Naram-Sin led an army to fight off the Amorites about 2200 BC. But after the collapse of the Akkadians, around 1900 BC, the Amorites succeeded in invading Mesopotamia. So by about 1700 BC their descendants created the Babylonian Empire. This empire centered on their capital city of Babylon. Babylon was the same place as the old city of Akkad, the old Akkadian capital.
When the Indo-European Hittites invaded Turkey, probably the Amorites were not officially conquered, but they did come under the influence of the Hittites and learned a lot from them. Mostly, the Amorites learned how to ride horses and how to use a war chariot for fighting.
The Amorites seem to have used this knowledge to attack Egypt, about 1700 BC. At first they probably attacked the parts of Syria that were under Egyptian control. When the Amorites won there, they continued on south along the Mediterranean coast through Lebanon and Israel into Egypt itself, where they seem to have controlled the mouth of the Nile (the area around Memphis) for a while. The Egyptians called them the Hyksos (HICK-soss), which just means the foreigners, the strangers.
With the start of the New Kingdom in Egypt, the Egyptians forced the Hyksos/Amorites to leave Egypt. The invaders went back to their own land in Syria and southern Turkey. We don’t know very much about them after that. The Christian Bible mentions them, placing them around 1200 to 700 BC. They seem to have eventually been absorbed by the Assyrian Empire.
Find Out About Mesopotamia: What Life Was Like in Ancient Sumer, Babylon and Assyria, by Lorna Oakes (2004).
Archaeology of the Land of the Bible: 10,000-586 B.C.E., by Amihai Mazar (1992). It’s only marginally about the Amorites, but there’s not much out there.
Ancient Near Eastern History and Culture, by William H. Stiebing (2002). Expensive, and hard to read, but it’s a good up to date account.