Originally Sumer was the southern portion of the lower half of what is now Iraq and Akkad was the upper portion. This map shows roughly where each was located. Note that "AKKAD" in capital letters refers to the region and "Akkad?" in lower case refers to the city of Akkad (also known as Agade), which hasn't been identified yet.

While both Sumer and Akkad shared the same culture and may have both initially spoken the Sumerian language, speakers of the Akkadian language slowly became more prominent. Initially the evidence for Akkadian consisted of Akkadian personal names, but full documents were written in Akkadian by about 2500 BC. As you'd expect from the name, Akkad appears to have been the region where Akkadian was first dominant.

Politically both regions consisted of a number city-states, which were united a number of times in the third millennium BC. The most prominent of these were by the Akkadian Empire, founded by Sargon around 2330 BC, and the Third Dynasty of Ur (AKA Ur III or the Neo-Sumerian Empire), founded by Ur-Nammu.

By the end of the third millenium BC and the fall of the Ur III dynasty, the Sumerian language was dead as a spoken language but remained as a literary and religious language (similar to Latin in the middle ages). The area was dominated by city-states again, many of which were ruled by Amorite speakers.

One of these city-states was Babylon, which under Hammurabi conquered most of Mesopotamia. His successors eventually lost most of this territory, and Babylon was taken over by the mysterious Kassites. While the Kassites were a foreign dynasty, Marc Van De Mieroop has said that it was under them that the the regions formerly known as Sumer and Akkad developed an identity as Babylonia.

Speaking of Van De Mieroops, his A History of the Ancient Near East is an excellent history of the region, although I imagine your professor has his or her own ideas.

In the northern portion of Iraq, Sumer and Akkad initially dominated but by the start of the second millennium BC the city-state of Ashur dominated and the region was known as Assyria.

tl;dr Sumer was the southern portion of the southern half of modern Iraq between the Tirgis and Euprhates, while Akkad was the northern portion. Eventually they became known as Babylon. Assyria was the northern portion of modern Iraq.