Sumerian was spoken in southern Iraq in the fourth and third millennia BCE, and it is the language of the earliest substantial writing anywhere. Cuneiform was developed to write it, beginning as an accounting notation and becoming a full script capable of recording literature, law and liturgy. Sumerian has no demonstrated relative.
Proposals connecting it to Dravidian, to Uralic, to Caucasian languages and to others have been made and none has been accepted, and the reason is partly that the language is genuinely isolated and partly that the comparative method has very little to work with when one member of a proposed pair is attested only in a script that under-represents its phonology. It is also typologically unlike its neighbours: agglutinative and ergative where Akkadian, the Semitic language that shared the same cities, is neither. The two were in contact for centuries, and each borrowed heavily from the other. By around 2000 BCE Sumerian had ceased to be anybody's mother tongue, which in most cases is the end of a language's story. Here it was not. The isolation is worth holding onto for a further reason. Sumerian stands at the head of the written record for the whole ancient Near East, and much of later Mesopotamian religion, written in Akkadian, is continuous with what it first recorded. The oldest religious literature anyone can read is therefore in a language with no known relatives, and nothing in its grammar can be checked against a sister tongue.