A cur is a dog that isn't very good — or is a mixed breed. If dogs understood English, they would be offended at being called a cur.
When you use the word cur, you're talking about a dog that's either a mutt, very unattractive, aggressive, or all three. The word can also be used as an insult for a person, especially a despicable man. In Old English, the word cur was originally cur-dog, which sometimes was used to describe any dog — good, bad, pedigreed or part Shepherd, part Collie, part whatever.