The purpose of a dog registry is to ensure dog and puppy owners of the sound nature of their new puppy as well as to validate the authenticity of a bloodline.
Dog Registries are comprised of organizations formed to compile a register of an official list of dogs within a specific dog breed whose parents are known to the registries and contain a "clean" bloodline. Most puppies are registered when they are born by their breeders. Registries also may compile a "stud book" used to reference male dogs within a specific breed.
Some kennel clubs maintain their own dog registries within their own group of members and affiliated clubs.
There are a number of puppy and dog registries you can list your puppy through.
AKC - American Kennel Club
APRI - American Pet Registry
UKC- United Kennel Club
WKC- World Kennel Club
WWKC- World Wide Kennel Club
Dog registration is not required but is a great idea to get your new puppy or adult dog registered if your breeder has not already done so. If you intent to show your dog, this should be the first thing you do. There are breed specific registries where you can dual register your dog as well. Many show dog owners have their dogs registered through a standard registry such as AKC as well as a secondary registry for their specific dog's breed.
Keep in mind that a registration does not insure the quality or soundness of a dog or dog breeder. It is more important to find the right dog for your needs and personality than it is to find a registered dog. Look into the meaning of each different dog registry listed above and learn more about the type of registration your new puppy may already have or your options if registering your own dog.