Caveat: I am not an Assyriologist, but I have studied the ancient Near East somewhat and can hopefully clarify these terms, It may be more helpful to think of these as geographical and lingustic designators rather than periods, especially since they are all very closely intertwined. Before we get started, [here is a reasonably usable map of Mesopotamia] (http://oi.uchicago.edu/OI/PROJ/NIP/PUB93/NSC/NSCFIG1.html); note that it is missing the city of Arbela, which is to the near southeast of Nineveh.

1. Sumerian refers to three things: A language isolate(meaning, not related to any other known language) that appears to have been spoken in Mesopotamia until somewhere betwee 2200-1800 BCE( to give the most generous range possible for when Sumerian died out as a spoken language; the actual timeframe is a topic of considerable debate because Sumerian stayed in use as a language of literature and culture long after it died out much like Latin did in the medieval and early modern west), the southern quarter or so of Mesopotamia(called "Sumer"), and the ethnic group which spoke Sumerian natively. What exactly this ethnic group was is a vexed question, not least because our present textual sources do not make it clear what people considered "ethnically Sumerian" and the whole project of finding the "Sumerian people" is to some extent an anachronistic project, but people still did and do it. When people speak of "Sumerian" history generally speaking they are referring to the pre-Akkadian period(before c. 2300) and the period of the "neo-Sumerian" city-states(from roughly 2100-1800 or so; all dates should be presumed approximate).

2. Akkadian refers to either the Akkadian language, a semitic language related to Hebrew, Arabic, Aramaic, and similar languages, a region roughly corresponding to the next quarter of Mesopotamia north of Sumer the native speakers of Akkadian(likewise difficult to pin down as an ethnic group, although to some extent in our early Cuneiform texts we can observe a gradient of more "Semitic" personal names in the northern range of the script and more "Sumerian" personal names in the southern range"), and the dynasty of Agade, founded by Sargon of Agade(often written Akkad, hence the name Akkadian).

3. Babylonia(or as it was called in antiquity Sumer and Akkad) is the southern half of Mesopotamia, named for the city of Babylon. Because this region contains the oldest and some of the most prominent cultural centers of Mesopotamia, Babylonian as an adjective is often used to refer to Mesopotamian culture of the second and first millennium more generally. Babylon was an important capital at several points in the second millennium BCE before being absorbed into the neo-Assyrian Empire; it then experienced a brief period of renewed independence after the fall of Assyria before finally being absorbed into the Persian Empire. Even when it was not independent, however, Babylon and the cities to the south of it like Nippur, Uruk, and Larsa were very important centers of Mesopotamian scholarship and culture. Babylonia as a whole is very flat and in antiquity was crisscrossed by a network of canals that its agriculture depended on; southern Sumer has historically been very marshy and in particular the southernmost cities of Sumer were historically very close to or on these near-littoral marshlands.

4. Assyria refers to the region north of Babylonia, defined more or less by the upper Tigris and Euphrates basins and centered in ancient times on the cities of Assur, Nimrud, Nineveh(now mostly in the suburbs of modern Mosul) , and Arbela(mostly under modern Erbil). It also refers to the state centered on this region that existed in one incarnation or another from roughly 1900 BCE to 612 BCE and which famously flourished into an empire that dominated all of Mesopotamia and much of the Near East generally in varying degrees from about the 9th to the 7th century. Assyria is somewhat hillier and more elevated; the Tigris in particular is faster which makes irrigation harder but the climate is more suitable to dry(non-irrigation-based farming), especially in the northern parts of Assyria. The Assyrian state also had in antiquity a long history of close trade ties with Anatolia(Modern Turkey) and Syria even before its campaigns to conquer the Syrian city-states, and along with Babylon it had cultural ties to Syrian cities like Mari and Ebla in the second millennum.

Assyria is also the region and term which has had the longest currency of the ones you mentioned; it remained as a designator for northern Mesopotamia even into the Sassanian period as Asoristan and remained in use as the name of the some modern eastern Christians of the region and their descendants who speak various Aramaic dialects and most of whom are members of the Assyrian Church of The East . Whatever continuity exists between modern Assyrians and the ancient Assyrians is highly contested; I won't get into it here except to note that it is very common for a lot of middle-eastern ethnic groups to try and claim continuity with highly varying degrees of plausibility to ancient ethnic groups and states as a way of promoting themselves in the present and legitimating their land claims, such as the attempts by Turkey to claim kinship with the Hittites, the attempts of Israel to use archaeology to justify historic Jewish claims on current Israeli territory or the various Palestinian claims of descent from Canaanites. But the examples could be multiplied ad nauseam.

Hopefully that overview was helpful! If you're interested in reading about Mesopotamian history in more detail, I would suggest reading Marc van der Mieroop's A History of the Ancient Near East as the most popular one-volume summary of Mesopotamian history.

To give you a tl;dr for /u/farquier who was very comprehensive, here's what I think you're looking for (note that I'm by no means an expert, this is from my 'archaeology of the Near East' course):

• Sumerian: refers to a common culture in southern Mesopotamia around 2900 - 2340 BCE. This was a period when city-states developed, secular authorities emerged.

• Akkadian: roughly 2340 - 2150 BCE, for the first time southern Mesopotamia would be unified under one man, Sargon or Sharu-ukin of Agade (which hasn't been located). Power moved from Sumerian speakers to Semitic speakers along with changes in political organizations, kingship, art but there seems to be some continuity between the two. This is the model (a unified Mesopotamia) that later kings will try to emulate, most won't succeed.

• Assyria: you have multiple periods, the 'Old Assyrians' period was around 2025 - 1809 BCE, 'Middle Assyria' from 1392 - 1056 BCE, and 'Neo-Assyrian' from 911 - 627 BCE. Obviously the city was around outside of those periods, these are just when it was prominent or a regional power. 'Old Assyria' became powerful through its trading networks (more of a merchant state) but was taken over by the Amorites under which their trade networks shrunk and then got eclipsed by Babylon. There was a resurgence under the 'Middle Assyrians' who became a great power competing for dominance with states like Babylon, the Hittites, Egypt, etc... They'll collapse (like most of those great states) at the end of the bronze age. I can't say anything about the Neo-Assyrians, haven't gotten to them in my studies.

• Like the Assyrians, Babylon had several periods when it became prominent. The most famous would have been under Hammurapi (or Hammurabi) in 1792 - 1750 BCE where he controlled all of Mesopotamia. However the empire began shrinking almost immediately after his death and his sons weren't able to hold it together. In 1595 BCE a foreign group called the Kassites took over and would hold power in Babylon for 576 years. They were hated as outsiders but brought wealth and stability, but not a huge military power. This period ends with the sack of Babylon by the Assyrians and Elamites in 1019 BCE.