Alrighty, first up 'Mesopotamian' refers to the Sumerian and Akkadian (Assyrian/Babylonian) peoples living in Mesopotamia (around the area of modern Iraq) that dominated the region for a period of 4200 years from the fourth millennium BC to proximately the 3rd century AD. Christianity began to take root among the Mesopotamians in the 1st century AD, and over the next 300 years the native religion largely died out.
So you might think the Romans and Egyptians had gods for everything, but the Mesopotamians pretty much have them both beat: historians have managed to count up 2100 different deities for almost freaking everything. Lots are just gods of specific regions and others we have no idea because we've just got their name written on a piece of pottery somewhere.
Although it mostly died out 1600-1700 years ago, Mesopotamian religion has still had an influence on the modern world because because Biblical mythology (Judaism/Christianity/Islam) stole bunches of stuff from them, like the Creation myth, the Garden of Eden, The Great Flood, Tower of Babel, Moses' origins, and the Ten Commandments. And of course the neopagans have picked up on the gods/belief systems of Mesopotamia as well. (The pagan movement of the ancient Mesopotamian religion being known as Kaldanism.)
According to the Mesopotamians, the world was created when the god Marduk killed the mother goddess Tiamat and used half her body to create the earth, and the other half to create both the paradise of šamû and the netherworld of irṣitu. (This can be found in the Epic of Creation, dated to 1200 BC.) The universe was considered to be a sphere with three levels of šamû (where the gods dwelt and where the stars existed) above the three levels of earth below it.
The gods were referred to us 'dingir' in the Sumerian language and ilu in the Akkadian language. They were seen as being very similar to humans: they drank and ate and got boozed up and did stupid things. They weren't really seen by the people as mystical beings but more like high-up masters who had to be obeyed and feared, not loved and adored. Among the gods there were ranks and levels that even they had to obey.
The Major Deities
Gods Adad/Ishkur - god of storms, venerated as a supreme power especially in Syria and Lebanon
Anu - One f the oldest gods of the Sumarians, Anu was the god of heaven, lord of constellations, king of gods who dwelt in the highest heavenly regions. It was believed that he had the power to judge those who had committed crimes, and that he had created the stars as soldiers to destroy the wicked.
Enki - god of crafts, mischief, water, seawater, lakewater. intelligence, and creation. His symbols included a goat and a fish which they later smooshed together which is where we get Capricorn. Prometheus Enki made mankind out of clay and his own blood and always remained close to them, and when Zeus Enlil decided he wanted to drown them all because their noise was keeping him up, it was Prometheus Enki who protected them and told Noah some dude to build a boat for his family and all the animals.
He also tries to seduce Inanna at a God-party and she's all "not tonight sweetie" and he's all "awww, come on" and gives her his Me, which is basically all the gifts of civilised life. In the morning when he's totally hungover he's all "servant! bring me my Me!" and the poor servant has to inform him that, uh, he totally gave it to the hot goddess last night when he was trying to get laid. And he's all fuuuuuuuuuck and has to send a bunch of demons to try and get it back off her.
But he obviously isn't too against Inanna later because when she gets stuck in hell, he's the only one who'll actually help her out. He also helps her out later when this guy rapes her. Enki, you might notice, is a pretty decent bloke all around, even to the ladies, which is especially impressive considering the Mesopotamian pantheon (and general society) were all about the patriarchy.
He also wears the skin of a carp. Because he can.
Enlil - Lord of the Storm, big kahuna of all the Meso regions, god of breath, wind, divine ruler of the Earth and its human inhabitants. (Which should make it totes okay that he tried to destroy them a million times over - that flood was just one of the times.) He was born from the exhausted breath of the god of the heavens after he finished boning his lady, the goddess of the sky.
Marduk - patron deity of Babylon who eventually became regarded as the head of the Babylonian pantheon. He was the god of water, vegetation, judgment, and magic.
Nabu - god of wisdom and writing. He rides around on a dragon and the planet Naboo is named after him, for realsies.
Nergal - god of plague, war, and the sun in its destructive capacity; later husband of Ereshkigal. He's a moody bitch with titles like "raging king" and the "furious one", fiery and destructive he was associated with the planet Mars and has connections to Ares.
Ninurta - champion of the gods, the epitome of youthful vigour, and god of agriculture. He slays whole bunches of monsters is a rather Herculean fashion.
Shamash/Utu - god of the sun, arbiter of justice and patron of travellers. He's associated with Saturn.
Sin/Nanna - god of the moon. Sin had a beard made of lapis lazuli and rode on a winged bull.
Tammuz/Dumuzi- god of food and vegetation, the consort of Ishtar who was confined to the underworld for half the year so that Ishtar could return above. He's your classic life-death-rebirth kinda guy.
Goddesses Ereshkigal - Queen of the underworld realm of Irkalla where everyone went, good or bad. (Sometimes she is also called Irkalla in the same way that the Greeks used 'Hades' interchangeably for both God and Place.) She was the sister of Inanna and the two did not get on particularly well. Or at all. In most stories she rules the underworld all on her ownsome, but in other myths she has a husband who is subordinate to her and that she was kidnapped away to the underworld and made its queen unwillingly.
They once invited her to an awesome heaven party, but Ereshkigal was terrified of the light and sent her servant to go in her place. The gods were well pissed off about this slight and they sent the god Nergal to punish her. He pulled her off her throne to decapitate her but her pleading made him stop and instead he married her. How utterly romantic. The Nergal connection is one that appears later in myth.
Inanna/Ishtar - Queen of Heaven, goddess of lust, sex, fertility, and war. The most prominent of the female deities, associated with the planet Venus and with lions and often pictured standing on the backs of two of them. (Lions, not Venuses.) Like Venus, Inanna seems unpredictable in her actions, being both the goddess of love and war, having both masculine and feminine qualities, and occasionally having temper tantrums. Inanna is the only deity who was able to travel into the underworld and return to the heavens again. She's a love goddess without the marriage part, being more related to affairs and sexy backstreet meetups, and would prowl the streets and taverns looking for sexy men to bone. She treats said lovers truly atrociously, but that's okay because her husband is a bit of a dick who was sitting around having a glorious old time while she was trapped in the underworld.
Nanshe - goddess of social justice, prophecy, fertility and fishing. She nurtured orphans, provided for widows, gave advice to those in debt, and took in refugees from war torn areas
Ninhursag - earth mother fertility goddess who often was depicted leading a lion cub on a chain awwww
GugalannaNamtar - god of death, minister/messenger of Ereshkigal, responsible for all diseases and pests.
Ningishzida - a god of the underworld who was sometimes depicted as a serpent with a human head and his symbol is the first example of the caduceus.
Goddesses Aya/Šherida - One of the oldest Sumerian gods, goddess of the rising sun, sexual love, and youth
Mamitu - goat-headed goddess of destiny, who decreed the fate of the new-borns, goddess of the oath, later a goddess of fate and a judge in the underworld
Nidaba - goddess of writing, learning and the harvest. She was the scribe of the gods and in charge of marking out regional borders.
Ninkasi -the goddess of beer. Her hymn is the earliest written recipe for beer and some people made a few years back and it turned out to actually rock.
Ninshubur - Queen of the East, messenger goddess and second-in-command to Inanna, associated with the planet Mercury and was probably at least partly the basis for Hermes
Tiamat - primordial goddess of the ocean and chaos monster.
Demigods and heroes
Adapa - a hero who unknowingly refused the gift of immortality
Enkidu - Enkidu embodies the wild or natural world, and though equal to Gilgamesh in strength and bearing, acts in some ways as an antithesis to the cultured, urban-bred warrior-king. Enkidu then becomes the king's constant companion and deeply beloved friend, accompanying him on adventures until he is stricken ill. The deep, tragic loss of Enkidu profoundly inspires in Gilgamesh a quest to escape death by obtaining godly immortality.
Gilgamesh - Gilgamesh is the central character in the Epic of Gilgamesh, the greatest surviving work of early Mesopotamian literature. Gilgamesh is a demigod of superhuman strength who built the city walls of Uruk to defend his people from external threats, and travelled to meet the sage Utnapishtim, who had survived the Great Deluge. He is usually described as two-thirds god and one third man. He had mega adventures all over the land and pissed of Ishtar but refusing to have sex with her.