1. A random-bred or mixed-breed dog. Usually used in a derogatory way.
2. A highly-specialized type of dog, usually bred for hunting. Cur dogs don't conform to a breed standard, as they aren't a true breed. They are bred solely based on their working ability and vary quite a bit in appearance. Not derogatory when used to describe this type of dog.
1. "What kind of dog is that?" "Oh, I don't know... just a cur."
2. "That's a nice cur dog you've got. Bet he's great with hogs."