I can't speak for other species, but for dogs, the AKC page on Finding Your Dog describes "responsible breeders as:
The American Kennel Club believes that breeding programs should be undertaken responsibly for the purpose of preserving breed characteristics and producing healthy, well-socialized puppies. Responsible breeders are expected to give careful consideration to health issues, temperament, and genetic screening, as well as to the individual care and placement of puppies in responsible homes.
They also say about discussions between you and the breeder (emphasis mine):
Your contact should be willing to answer all of your questions fully. He or she should also ask questions about you, your lifestyle, and your family. If a contact doesn't respond to your inquiries, or doesn't show any interest in the life the dog will lead after it leaves his or her premises, you may want to look elsewhere.
A good breeder cares about the animals they raise. They want to be sure that it has a good life with you, and vice verse. They also care about raising good animals, so they should be able to provide you with details of health screenings above and beyond normal vet visits such as CERF (eyes) or OFA (orthopedic) screenings. They may want to visit your home to see the environment the puppy will live in, or have you visit them to meet the puppy on its home ground.
A good breeder will also keep good records about the animals that they are breeding. They won't be bred until fully physically mature, and they won't be bred after a certain age (I've heard between 6 and 8 years for different breeds). If it's an AKC-recognized breed, they should be able to provide AKC paperwork for the puppy.
Expect to sign a contract when you purchase the puppy. If it's not sold as a show dog, you can expect a spay/neuter clause. You could also see a clause about returning the dog if he or she doesn't work out at your home.
A puppy mill cares very little about any of this; dogs are often bred as soon as they reach sexual maturity, and are bred on every heat until they die. They are often kept in unsanitary conditions with little opportunity for exercise or socializing. All they care about is getting your check and handing over a dog.