Mesopotamia is not a nation or ancient civilization, but a strip of land in Iraq between two rivers: the Tigris and Euphrates. The most important (and only) things you should know about ancient Mesopotamia are that this was a place were one of the first human civilizations rose along with basic but essential things such as writing and the wheel. Despite this, here's more information about other civilizations and empires that followed the inventors of the wheel (also known as the Sumerians).
The Sumerians lived in the southernmost part of Mesopotamia and established the ancient civilization of Sumer (kind of sounds strange). The Sumerians had inhabited the region since 4500 B.C, but cities began to rise around 3600 B.C as a result of an increase in agriculture. After the rise of the Sumerian civilization, inventions such as the wheel and (arguably) writing developed. One architectural invention of the Sumerians was the ziggurat. This pyramid-like temple was made of mud bricks and was adopted by other Mesopotamian nations after the Sumerians. The fall Sumerian civilization began around 1750 B.C, when attacks by Semitic tribes eventually led to migration of the Sumerians. Eventually, the Sumerians were absorbed into the Semitic population.
Right, top- A ziggurat located in the Sumerian city of Ur.
Right, bottom- Statues of the Sumerians.
Babylonia & King Hammurabi
After the Sumerians, the city Babylon began to become a strong city-state. By 1750 B.C, king Hammurabi had made what historians call the Babylonian Empire (with Babylon as its capital, of course) into the most powerful nation in southern Mesopotamia. King Hammurabi was not just famous for establishing the "Babylonian Empire" but also for his code of laws. The Code of Hammurabi (or as I call it, the Hammurabi Code) was used in business, public, and family matters (such as marriage/divorce). These 282 laws are known for establishing the principle of "an eye for an eye" and were written in a "if, then" fashion. They were also known for sometimes been harsh. An example can be, if you kill someone's wife, then your wife would be put to death. Hammurabi's Babylonia survived three more centuries until around the 14th century B.C, when a group of people in northern Mesopotamia began to supersede Babylonia.
Left, bottom- Part of Hammurabi's Code.
picture source- https//:www.wikipedia.org
The Assyrians and Their Empire
These people from the mountains to the north came from the city of Assur, which is why they were called the Assyrians. They were able to cause the downfall of Babylonia for several reasons. One of the main reasons why the Assyrians superseded the Babylonians was because of their brutality. The Assyrians were described as semi-nomadic, looking for territory to take over and settle. The Assyrians also had some military advantages, including the horse and the chariot. This lead to the creation of the Assyrian Empire, one of the biggest and first empires of the ancient world (and all history). The Assyrians were even able to expand their empire past Mesopotamia into other regions like northern Egypt! The giant size of the empire however, caused a weakening on the Assyrian state. The final blow eventually came around 700 B.C, when Chaldean people (from the coast of the Persian Gulf) were able to force the Assyrians out of their territory. This lead to a slow decline of the original Assyrian Empire as the Chaldeans began to rise.
Right, bottom- Depiction of an Assyrian chariot.
picture source- www.pinterest.com
Epilogue: Other Empires
I am writing this section for one purpose.... to get rid of confusion. I'm sure that if you Google the names of some empires (Assyrian Empire for examples), you'll get two results. You might get the first Assyrian Empire (which I wrote about above) or the Neo-Assyrian Empire. There are other similar empires with confusing names like the Neo-Babylonian Empire (which is another name for the Chaldean Empire from above). Sometimes the names are mixed together but you just need to find a way to make sure you're talking of the right empire. I just hope that this helps clear things out.
Left- This is the confusion I refer to: The Babylonian Empire and the Neo-Babylonian Empire (which is which?).
picture source- history-world.org