Backyard breeders also refer to those people who do put a little thought, time and effort into the breeding, but it's limited towards maybe buying a better quality food while she's pregnant and making sure they liked the looks of the stud dog they picked. They wanted a litter of puppies so their kids could "experience the miracle of birth" and have some rowdy puppies to play with for awhile, and hopefully the parents would be able to make a buck or two in the selling. These are generally the puppies you see in the papers. "AKC yellow lab, big, friendly, 250." The AKC title doesn't mean much anymore in advertising, because just about anyone can claim an AKC and most of the general public doesn't know enough to demand papers from both parents, etc.
Puppy mills are purely profit driven. They pick out the top 5-20 most popular breeds of the time, find a handful of b*tches and at least one corresponding stud dog. Then they breed and breed and breed, generally selling the puppies at too early of an age, with no thought to puppy care, b*tch maintenence or even proper housing. The puppies are sold to either pet stores, who mark up the price of the puppy, or to on-line pet stores. These puppies are generally the ones that have a tremendous amount of health problems, many of which stem from being weaned too quickly because the owners of them mill felt the need to breed the b*tch again right away. Watch Animal Cops sometime, you'll see a full-fledged puppy mill sooner or later. Most of the breeding stock is in such terrible shape from being constantly keep pregnant or nursing.
Health testing generally covers OFA--I forget what the acronym stands for, but it's the hip dysplasia test; known genetic eye and back problems and full or partial dentition (do the parents have a full set of teeth or not) and certain cancers. Like Athe pointed out, usually careful screening will prevent an unhealthy litter, but not always. Good breeders will give a health guarantee on their pups until a specified date, sometimes anywhere from six months to over a year. The really good ones will offer a replacement pup should something like hip dysplasia crop up in your dog anyway. I know a guy in the exact situation: his dog got HD at a really young age so the breeder is sending him another pup for free. He can keep his first dog as well.