The American Kennel Club (AKC), and the United Kennel Club (UKC) are the most widely used registries of pure bred dogs in the United States. These organizations classify dogs based on the purpose for which each breed was developed. There are seven categories used by the AKC and eight by the UKC that describe over 150 breeds of dogs. I've combined the AKC and UKC classes and added 2 more below. At least 150 more breeds are recognized worldwide and can be classified in the same manner.

Classifying dogs by breed purpose is a sound method for evaluating potential behavior. Mixed breeds can be classified with the breed or breeds they most resemble. Even though every dog is an individual and will be affected by early environment and training, if you truly want a retriever or hunting dog, it is important to start with a breed that has generations of expertise bred into him. Training will accentuate what comes naturally. You can also choose a mixed breed that has the physical characteristics common to the group of dogs you prefer, such as hanging ears and a round medium snout for a hunting dog. If you aren’t particular about specific training, choose a dog likely to adapt best to your family routine in energy level, size, and grooming needs. 

GROUP I: Hounds (AKC)
SightHounds (UKC)
Dogs bred to hunt by sight (Greyhounds, Whippets) are fast runners and enjoy chasing small animals though they are usually quiet when indoors. They were bred to work without direct supervision so they are more comfortable spending time alone then some other types of dogs. They are often aloof with strangers. They are long legged with deep chests, long noses and small ears. Most have short hair but the Afghan has long silky hair that requires considerable grooming. 
Scenthounds (UKC)
Dogs bred to hunt by scent (Beagles, Bassets, Coonhounds) enjoy trailing and chasing prey and are vocal about it. They were bred to live in groups so tend to get along well with other animals. They come in all sizes and have long ears. Most have short or wiry coats that require minimum care.

GROUP II: Sporting Dogs (AKC), Gun Dogs (UKC)
Sporting dogs (pointers, setters, spaniels) were bred to retrieve, swim and work all day with a hunter or farmer so they prefer to have company most of the time. They also have enough energy to go jogging and camping. If not kept active they can become overweight. They like to carry things in their mouths. They come in all sizes with soft, hanging ears. Many have their tails surgically shortened, called "docked" tails. They come in all coats including straight and short, wiry or long and silky.

GROUP III: Terriers (AKC, UKC)
Terriers (Scotties, Airedales, Bedlingtons) were bred to cleverly discover the hiding places of their prey--rodents. They are fearless and persistent and like to dig. Their pluckiness sometimes gets them into trouble with larger dogs. They come in all sizes. Most have wiry coats and short upright ears. Some have surgically shortened ears, called "cropped" ears, in addition to docked tails.

Group IV: Working Dogs (AKC)
Guard Dogs (UKC)
Guard dogs (Mastiffs, Dobermans, Beaucerons) are bred to work alone protecting territory or work with law enforcement officers. Guard dogs are suspicious of anything unfamiliar. These breeds usually form a close bond with one or two people although they can be very trustworthy guardians of children. Rescue dogs (Newfoundlands, St. Bernards) are also bred to work in conjunction with a handler finding people. Rescue dogs are much more trusting of strangers than Guard dogs. 
These dogs are all large and come in all coat types, some with cropped ears and docked tails.

Northern Dogs (UKC)
The Northern dogs (Huskies, Samoyed, Spitz) are also working dogs that pull sleds and hunt. They are not as territorial as guard dogs. They can be more predatory than other breeds and are comfortable spending time alone. In fact, Northern dogs often take off on their own if allowed to escape. They usually have a double coat with straight hair over a fluffy undercoat that sheds twice each year. Most northern breeds have upright ears and curled fluffy tails. 

GROUP V: Herding Dogs (AKC, UKC)
Livestock Drivers (PBC)
These dogs (collies, shepherds) were bred to help farmers move herds of livestock. They do so by chasing, bumping into and nipping at the members of the herd that stray or lag. They are always alert and will be bored without interesting training. These dogs prefer to have company of other animals if possible. They have upright ears and some are born with no tails.
Livestock Guards (PBC)
These dogs (Great Pyrenees, Pulik, Komondor) are independent and can be suspicious of strangers. They are bred to protect herds of animals from predatory wildlife such as coyotes and wolves. Breeders often take puppies at a very young age and raise them with the herd of animals they will live with. These are large dogs that are comfortable resting with the herd all day and traveling distances over long periods of time. Many of these breeds have unusually long coats. 

GROUP VI: Companion Dogs (UKC)
Toy (AKC)
These dogs (Maltese, Shih Tsu, Chihuahua) were bred to be pampered companions. Those with long coats need extra grooming care. They are very intelligent and sensitive and quickly communicate their wishes to their caretakers, sometimes becoming quite demanding. They enjoy attention and will learn any tasks that earns it. The toy breeds are small, come in all coat types, and some have docked tails.
Non Sporting (AKC)
All dogs whose original purpose is outdated are classified here. Dalmatians were bred to keep carriage horses company, Chow Chows to guard palaces (they also were used as a source of food, so forgive them if they are a little suspicious of strangers) and Bull dogs to catch and hold animals for a butcher or farmer.  

Pariah or Feral Dogs (not necessarily purebred):
The UKC recognizes some of these breeds (Carolina, Canaan, New Guinea Singing)) as sight hounds. These dogs developed around human settlements for centuries and some can trace their ancestors reliably enough to be considered pure breds. These dogs developed without direct selective breeding by people and are truly multifunctioning dogs similar to the feral Dingo of Australia. Most of them have short to medium coats, upright ears, and are various shades of reddish brown, black or white. 

Fighting Dogs (not necessarily purebred): 
A group of dogs (Dogos, Canary Island, Tosa Inu) gaining popularity are dogs bred to fight with other dogs. The smaller fighters are terriers mixed with mastiffs.  Pit dogs in the past were bred to be aggressive to other dogs or animals but easily handled by people. Since dog fighting has been illegal in US and United Kingdom for over 100 years, many breeders of these dogs are operating outside the law. Some are selecting mastiffs bred to hunt large predatory animals such as cougars or bred to track escaped prisoners in order to create a dog that is aggressive toward and capable of injuring people. Due to their size, they pose a problem for animal shelters and animal control agencies. See Pit Bulls for more information. 

Discover the purpose of the breed you are considering to determine if the breed is right for you and be prepared to be on a waiting list for a puppy. NO REPUTABLE BREEDER SELLS IN A STORE ON THE INTERNET OR THROUGH A NEWSPAPER CLASSIFIED AD. Reputable breeders wait until they have a list of interested buyers to breed.