As a new cur owner, I'll take a shot at answering. Until about five months ago, I thought cur just meant mutt. It turns they are actually bred selectively.
My dog is a Mountain Cur, which is apparently a UKC recognized breed. I know that Black-mouth Curs are another common variety - I think they are also a breed. There seems to be a bunch of different types of curs bred; I'm not sure how many are actually "breeds".
They are multi-purpose dogs that were commonly used by settlers, particularly in the southern US. They do lots of hunting (squirrels, raccoon, boars, and even mountain lions). Some lines herd, and some are watch dogs. Old Yeller (in the book, not the movie) was a cur.
Although I certainly didn't seek out a cur to adopt, I'm madly in love with mine. She spends all day working with me at the horse farm I manage, is unfailing good-natured and polite with all people, loves hanging out on the porch in the evening, and is going to clean up in agility in another year or two. I don't know if she's typical of the breed, but I can't say enough good things about her.