The Ziggurat at Warka
Sumerian architecture is probably the oldest serious architecture (not just building houses and barns) in the world. People living in the area between the Tigris and the Euphrates rivers (modern Iraq) began to build really big, substantial buildings about 3500 BC. Because there's practically no building stone in this area, but there's lots of clay, Sumerian architects built their buildings out of mud-brick or fired brick.
This was so early that the architects didn't know how to make a big building stay up if it was hollow inside, so the first big buildings are solid - like a sand-castle - rather than really useful as buildings. They're more like artificial hills. This is the same as the Egyptian pyramids, which were built just a little later and are also pretty much solid inside.
Mostly what they built was huge staircases of mud-brick which are called ziggurats. Each little city-state would build its own ziggurat, partly to please the gods and partly to show how powerful the town was. On top of each ziggurat, there was a small temple to Ishtar or Anu or another Mesopotamian god.
The Sumerians also built town walls around their towns, which were also built mainly out of mud-brick, and which could also be solid. (In fact fortification walls pretty much have to be solid!).
King's palace (Mari, ca. 1800 BC)
The Sumerians in each city-state built palaces for their kings, too. These palaces weren't just to live in; they were also storehouses for wheat and barley and cloth and all kinds of things that the kings collected as taxes. And of course not only the king but also his whole family and many slaves lived in these palaces. The palaces were also made out of mud-brick.
Saddam Hussein's reconstruction of the ziggurat at Ur
Mud-brick buildings like these would look pretty boring if they just had straight walls, all brown. So the builders made them look more interesting by creating areas of dark and light on their walls - the walls went in and out at regular intervals, making a sort of pattern of shadows.
Sumerian reed hut
Ordinary people didn't live in palaces, though - they lived in simple reed huts like this one. They sat and slept on the ground, on woven reed mats, or on mud-brick platforms with woven reed mats on them. They cooked over a firepit in the middle of their hut, or in a mud oven in the courtyard outside their house.
To find out more about Sumerian architecture, check out these books from Amazon or from your library:
The Art and Architecture of the Ancient Orient, by Henri Frankfort (5th edition 1997). The standard for college art history classes.