Although they are sometimes well-meaning and really care for their dogs, they do not do genetic health testing or temperment testing on their dogs. They do not take the time to find a suitable mate for their dogs. They do not participate in showing or other events to help verify that their dogs are good examples of their breed. Sometimes these people can be hard to spot - but make sure you get proof of successful genetic test outcomes on the sire and dam of your prospective pup. A pup from a backyard breeder has a much higher likelyhood of developing health and temperment problems later on, then one from a responsible breeder does.
Many times there are puppies and kittens available through shelters or rescues.
But if you cannot find what you're looking for, please do the research and find a responsible breeder. Another important thing to consider, especially with dogs, is what type of dog to get. Each breed has its own typical temperment. Many of the dogs in shelters today are there because its owners didn't research their breed and find out what to expect from their new dog. Even if you are planning to get a mixed-breed dog, some research will help you predict possible traits based on the dog's breeds if they are known.