The Babylonian Empire was actually not the first "great civilization" in Mesopotamia, and really wasn't even the first empire in that region. The first "great civilization" was the Sumerians, who invented the first writing, a writing system (cuneiform) used for thousands of years after they were gone. They were one of the two prominent cultures of the region, the other being the Akkadians, who used the Sumerian cuneiform system to write their own Akkadian language. The Sumerians were mainly based in southern Mesopotamia, and rose to regional prominence in the Uruk period, in the 4th millennium BC. The Akkadians arose later, in the 3rd millennium BC and were based mainly in the northern part of Mesopotamia.

The Akkadians conquered Sumeria approximately 2270 BCE, though Sumerian culture was still in force and Sumerian remained in use as a sacred language. The Ur III period saw a resurgence of Sumerian authority in the south, until climatic problems stemming from rising salinity in the soil from poor management of irrigation and high aridity destroyed the Sumerian economy, leading to a massive population shift to the North, and the Akkadian empire surged to prominence for a couple centuries. They too succumbed to economic issues stemming from climatic problems such as a massive drought, which may have also had a hand in the collapse of the Egyptian Old Kingdom. This drought brought on lots of conflict with neighbors and vassal states, because if you can't grow your food, you have to steal it. For a time Mesopotamia was full of small groups attempting to establish themselves, and here is where the Babylonians begin, as the Amorites come in and take over where Sumer used to be, adopting the Akkadian language, and eventually became Babylonia. The Babylonians were one of a number of competing groups for prominence in the region, and eventually won out under Hammurabi to become the major player in all of Mesopotamia, creating the first Babylonian Empire, which competed with the remains of the Akkadians in the north of Mesopotamia for control of that region. By this time, the Akkadians had developed into what we call the Old Assyrian Empire. Hammurabi's successors couldn't hold on to the rest of his empire and it quickly disintegrated after his death.

Babylon then fell under the rule of the Kassite dynasty, and then other small groups such as the Elamites and the Second Dynasty of Isin, until the Assyrians conquered the place in 911 BC.

The Asyrians had their own issues, but eventually in 911 they had enough power to finally conquer Babylon and even expand their influence far and wide. This is often considered the first true empire, constantly expanding from modern Iraq into modern Turkey, against the Hittites and over to Egypt, conquering ancient Israel and Judah as described in the Hebrew Bible. Areas that were previously only nominally under Assyrian rule (basically they would accept that Assyria was over them, maybe pay taxes when they absolutely HAD to, and otherwise were autonomous) were conquered and firmly placed under direct Assyrian control, installing their own puppet kings in areas they conquered and displacing entire populations to prevent uprisings and rebellions.

Finally, in 627 BC, at the death of Ashurbanipal, Assyria descended into a brutal period of civil war and revolution, which lost a lot of the expanded territory to revolutions and independence. One of these areas was Babylon, which was seized by a Chaldean named Nabopolassar, who, after allying with a few of the other vassal kings who had shrugged off Assyrian rule, including the Medes, Pesians, and Parthians, and launched a surprise attack on Assyria in 615 BC, and extended this assault, culminating in a massive battle at Carchemish, in Northern Syria, against the Assyrians and their Egyptian allies in 605 BC by Nabopolassar's son and successor Nebuchadnezzar II, resulting in the final nail in the coffin for Assyria.

And now, we finally get to the Babylonian empire that you are probably familiar with, under Nebuchadnezzar II who conquered Israel and Judah again and initiated the Babylonian Exile.

This empire lasted for only a few generations before the final king, Nabonidus. He was just a terrible politician. He left the rule of the empire to his son, and went off to fight for long periods of time. He had terrible relations with the powerful priesthood.

The Persian empire, led by Cyrus the Great, had been quietly rising, and finally conquered Babylon by the ingenious technique of diverting the Euphrates River and entering the city through its dry bed, with the assistance of Nabonidus' political enemies.

So as we can see from this extremely shallow overview of Mesopotamian history, there are a large number of reasons why empires rise and fall. Many times, especially early on, mismanagement of the climate and economy played a huge role in the collapse of civilizations. In other periods, you have strong kings creating vast empires, but when they are gone their successors are not strong enough to hold it, as internal politics may take up attention and lead to civil war. External forces occasionally come in and conquer these internally weakened empires.

Some sources for this overview include the always excellent A history of the ancient Near East, ca. 3000-323 BC by Mark Van de Mieroop, 2004 and Ancient Iraq by Georges Roux, 3rd ed. 1992