About the same time as the first Pharaohs were coming to power in Egypt, around 3100 BC, a group of people known as the Sumerians were living in the area between the Tigris and the Euphrates rivers, in West Asia (modern Iraq). The Sumerians themselves believed that they had come to the Euphrates from somewhere else, probably in boats from the Persian Gulf, and archaeology shows that it is true that their first cities were near the Persian Gulf, though they later spread out northward. But most people today think the Sumerians were actually native to Iraq.
To their east, in modern Iran, the Sumerians had the people of Elam (the Elamites), who they both traded with and fought with, at different times. To their west, the Sumerians had the Amorites, who spoke a Semitic language (related to modern Hebrew and Arabic).
The Sumerians seems to have begun to use writing by around 3000 BC, which allows us to know more about them, though we still can't really read their earliest writing. The Sumerians also used bronze. Sumer may be the first place where tin was added to copper to make bronze, maybe because there is so little stone there for tools.
They built big temples, called ziggurats, and mud-brick walls around their cities. They began big irrigation projects, digging canals and ditches to bring water from the Tigris and the Euphrates to the land between them so people could grow food there. In this way more people could live in the same amount of land.
The Sumerians told a story that all this happened "before
the flood". In the Sumerian version there were seven kings
before the flood, who had seven wise men (or half-gods) who helped them.
The first of these wise men was Uanna-Adapa, whose name is Adam
in the Bible. Then, the Sumerian texts and the Bible
agree, the gods were angry with men and sent a great flood which destroyed
everything, and only a few men survived to rebuild. The Sumerian
texts place this event around 2900 BC.
Archaeologists have looked for evidence of this flood in Sumer, and although it does not appear everywhere, it does seem that there was a serious flood in one area about this time.
The Sumerians, by Elaine Landau (1997). Despite the bad Amazon rating, this is a good solid introduction to the Sumerians, with an explanation of prehistory at the beginning for context. Pictures of ancient stuff, and good maps.
Sumer and the Sumerians, by Harriet Crawford (1991). A college-level introduction, with a lot of archaeological material and daily life information.
Find Out About Mesopotamia: What Life Was Like in Ancient Sumer, Babylon and Assyria, by Lorna Oakes (2004).
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.