Mesopotamia is the area of the Tigris/Euphrates rivers system, Sumer was one of the many civilizations that inhabited that area.
"Sumer was an ancient civilization and historical region in southern Mesopotamia, modern-day southern Iraq and Kuwait, during the Chalcolithic and Early Bronze Age. Although the earliest forms of writing in the region do not go back much further than c. 3500 BC, modern historians have suggested that Sumer was first permanently settled between c. 5500 and 4000 BC by a non-Semitic people who may or may not have spoken the Sumerian language (pointing to the names of cities, rivers, basic occupations, etc. as evidence). These conjectured, prehistoric people are now called "proto-Euphrateans" or "Ubaidians", and are theorized to have evolved from the Samarra culture of northern Mesopotamia (Assyria).
Sumerologist Samuel Noah Kramer asserts "No people has contributed more to the culture of mankind than the Sumerians" and yet it is only comparatively recently that we have built up a knowledge of the existence of this ancient culture."
https://en.wikipedia.org/wiki/Sumer
"Mesopotamia is a name for the area of the Tigris-Euphrates river system, corresponding to modern-day Iraq, Kuwait, the northeastern section of Syria and to a much lesser extent southeastern Turkey and smaller parts of southwestern Iran.
Widely considered to be the cradle of civilization in the West, Bronze Age Mesopotamia included Sumer and the Akkadian, Babylonian, and Assyrian empires, all native to the territory of modern-day Iraq. In the Iron Age, it was controlled by the Neo-Assyrian and Neo-Babylonian empires. The indigenous Sumerians and Akkadians (including Assyrians and Babylonians) dominated Mesopotamia from the beginning of written history (c. 3100 BC) to the fall of Babylon in 539 BC, when it was conquered by the Achaemenid Empire. It fell to Alexander the Great in 332 BC, and after his death, it became part of the Greek Seleucid Empire."
https://en.wikipedia.org/wiki/Mesopotamia
