The Anunnaki appear throughout Sumerian, Akkadian and Babylonian religious literature — some of the oldest written texts on earth, dating from approximately 2600 BCE though encoding traditions considerably older. The word Anunnaki (or Anunna) in the original Sumerian means something like "princely offspring" or "offspring of Anu" — Anu being the sky god, the highest deity in the Sumerian pantheon. They are the great gods collectively — the divine assembly that governs the cosmos, meets to decide human fate and is petitioned through ritual and prayer.
In the Sumerian tradition, the Anunnaki are divided between those who dwell in heaven (Anunnaki) and those who dwell in the underworld (Igigi — though this distinction is inconsistent across texts). They are not a race of beings from another planet — they are the Mesopotamian pantheon, the divine powers that govern natural forces, human institutions and cosmic order. Enlil governs wind and kingship. Enki governs wisdom and fresh water. Inanna governs love and war. Nanna governs the moon. Utu governs the sun. They are structurally similar to the Greek Olympians — divine personalities with specific domains, complex relationships and very human-like motivations including jealousy, ambition and desire.
The Sumerian understanding of the relationship between gods and humans is distinctive and important: humans were created to do the labour that the lesser gods (Igigi) refused to perform. The creation of humanity is explicitly described as a solution to divine labour unrest — the gods needed workers to farm the earth and feed them through sacrifice. This is not flattering to human dignity, but it is theologically honest about the function of human beings in the Mesopotamian cosmological system. We were made to serve, to feed the gods through offerings and to maintain the cosmic order through ritual. The relationship is transactional but also genuinely reciprocal — the gods, for their part, were expected to protect and provide for the humans who served them.