Kur was an underworld deity in the mythology of Sumer,  or the monstrous creature that roughly corresponded to the Babylonian Tiamat and the Hebrew Leviathan.  Kur lived in the empty space between the primal sea and the earth's crust. 
In the Sumerian creation myth, Kur stole a goddess immediately after the creation of the world:
After An had carried of heaven,
After Enlil had carried off earth,
After Ereshikgal had been carried off into Kur as its prize;
After he had set sail, after he had set sail,
After the father of Kur had set sail,
After Enki for Kur had set sail 
Against the king the small ones it hurled,
Against Enki, the large ones it hurled;
Its small ones, stones of the hand,
Its large ones, stones of reeds,
The keel of the boat of Enki,
In battle, like the attacking storm, overwhelm;
Against the king, the water at the head of the boat,
Like a wolf devours,
Against Enki, the water at the rear of the boat,
Like a lion strikes down.
-- The Sumerian Creation Myth 
Kur carried away the goddess Ereshkigal (counterpart to Persephone) to the nether world.  In response, the water deity, Enki, sailed to the nether world to attack Kur and avenge this theft. Kur hurled stones against the keel of Enki's boat as the primeval waters attacked the ship from all sides.  Enki overpowered the dragon and returned victorious. 
This particular myth of Kur is found in a prologue to Gilgamesh, Enkidu, and the Nether World as opposed to its own tablet.  As such, the prologue doesn't complete the story; it merely recounts the battle in a brief passage as part of the introduction to another. It is clear, however, from Enki's epithets such as 'Lord of the Abyss', that he was victorious in his endeavor. 
The Babylonian Epic of Creation centered largely around the slaying of the dragon Tiamat, but this myth was inscribed in Akkadian, a Semitic language, dating back to the first millennium B.C.,  which is more than one millennium later than the Sumerian literary inscriptions. Clearly the Sumerian dragon-slaying epic influenced the epics of later groups, including the Babylonians and the Akkadians.  The dragon-slaying theme was already an important motif in Sumer in the third millennium B.C. 
Etymology of Kur
The Sumerian word kur corresponds to several concepts and meanings developed over time. One of the primary meanings of kur is 'mountain,' which likely influenced the word's later translation into 'foreign land,' and then again later simply into 'land.'  Sumer itself can be described as kur-gal, 'great land.' 
The cosmic concept for the word kur, which can be identified with ki-gal, or 'great below,' which roughly translates to 'nether world.'  Thus, the cosmic meaning of kur is the empty space  between the primal sea and the earth's crust.  It is likely that this is the namesake for the monstrous dragon that dwelled at the bottom of this 'great below.' 
There are three great Sumerian myths of powerful entities overcoming, or slaying, Kur.  However, given the complexity of the Sumerian word kur, some myths and legends that utilize it refer to a mountain or foreign land or to the nether world or to the dragon that lived in the nether world.  Therefore, it is important to distinguish which entity is being referred to in a given myth.
Disambiguation of Kur
For example, Kramer attributes the name Kur to the demonic monster, which Kramer interprets as a draconic serpent, defeated by the mythic hero Ninurta, a prototype of Babylonian deity Marduk.  This myth comes from an epic over six hundred lines long,  roughly titled The Feats and Exploits of Ninurta. 
Ninurta's personified weapon, Sharur  (alternatively spelled Char-ur),  encouraged him to attack the demon, extolling and encouraging the hero to action.  However, others name this monstrous entity the Asag, who is thus described in the myth: The earth goddess Ki "bore Anu [the sky god] a warrior who knows no fear."  Asag was said to have produced monstrous offspring with Kur.  Ninurta destroyed the monster in battle. However, with Kur dead, there was nothing holding the primeval waters back, so they rose up and caused calamity to the land, making it so that no fresh water could reach the fields or gardens, resulting in famine.  In the end, however, Ninurta destroyed the demon Asag (or the dragon Kur) and set right all the imbalances of the world created by the monster's death: 
What had been scattered, [Ninurta] gathered,
What by Kur had been dissipated,
He guided and hurled into the Tigris,
The high waters it pours over the farmland.
-- The Feats and Exploits of Ninurta 
Are Asag and Kur synonymous, both referencing a dragon? Or are they different entities attributed as the dark force overcome by Ninurta? Owing to the age of Sumerian literature, and its predecessors adapting the myths, this remains uncertain.
Another disambiguation should be made between Kur the dragon and the myth of Inanna and Ebih.  Inanna was the Sumerian goddess of love, battle, and strife and among her many epithets was the 'destroyer of Kur.' In this myth, Kur is also called 'mountain Ebih,' which is a district northeast of Sumer. 
Therefore, Inanna overcame Kur, but in this myth, Kur is an inimical land, not a dragon. 
Kur was an enormous serpent, or snake-like dragon, living in the bottom of the 'great below,' where Kur kept contact with the primeval waters.