Kish was a city-state in ancient Mesopotamia. During the early history of the region (c. 3500 – 2334 BCE), city-states were ruled by kings, thus the king of Kish was the king of the city-state of Kish. Sounds simple, right?

Unfortunately, it's not simple. Because the term appears in a bunch of very old historical documents that make no sense if that is the only definition of the term. There are kings from places other than Kish, such as En-mebaragesi of Tutub/Khafajah (Diyala), claiming to be the “King of Kish,” when there is no other indication that the individual had ever been to Kish, let alone ruled there. So when you put all the historical documents together, it looks like the term became a name for the dominant king of southern Mesopotamia at some point. We don't have any details about what kind of powers the title may have possessed, but the later Lagash-Umma border conflict (c. 2350 BCE) makes clear that the person served as a kind of supreme authority or judge in conflicts in the region. In short, it was a term for some kind of early supreme king in ancient Mesopotamia at some points between 3500 and 2500 BCE.

Now just to make it more interesting, it turns out that the cuneiform sign KISH, which is originally a Sumerian (probably) word, happens to be very similar to the Akkadian word “totality (of the inhabited world)” kisshatum for reasons that are not entirely clear. A major consequence of this is that when the words “King of Kish” (Ak. shar kisshati) are written in Akkadian, it can mean “king of (the city of) Kish” or it can mean “King of Everything.” By the late Third Millennium BCE, the title “King of Everything” had become one of the primary titles of the dominant kings who ruled over a large region containing several cities.

The term fell into disuse in the early Second Millennium, but later terms, such as “King of Sumer and Akkad” (shar Sumeri u Akkadi) and the later, “king of Babylonia” (shar Babili) had very similar connotations.

TL:DR: It means “hegemonic king of lower Mesopotamia.”