Cur as slang refers to a type of random-bred, or mixed-breed dog

Mixed-breed dog

A mixed-breed dog, also known as a mutt or mongrel, is a dog whose ancestry is generally unknown and that has characteristics of two or more types of breeds. A mixed-breed may be a cross-breed dog, a random-bred dog, or a descendant of feral or pariah dog populations...

. This article deals with Cur as a breed.
The derivation of the word "cur" dates from the 13th century. It is thought to be short for the Middle English "curdogge", which derives from the word "curren", meaning "to growl". According to the Dictionary of True Etymologies the original root of the word may be Germanic

Germanic languages

The Germanic languages constitute a sub-branch of the Indo-European language family. The common ancestor of all of the languages in this branch is called Proto-Germanic , which was spoken in approximately the mid-1st millennium BC in Iron Age northern Europe...

, possibly from the Old Norse

Old Norse

Old Norse is a North Germanic language that was spoken by inhabitants of Scandinavia and inhabitants of their overseas settlements during the Viking Age, until about 1300....

 "kurra" meaning "to grumble".