Dogs have been man's best friend for all of recorded history. In fact, that's what Canis familiaris was designed to be. From guarding to hunting, herding to retrieving, sporting to companionship; dogs were bred by their human owners with specific purposes in mind.

Kennel clubs throughout the world categorize dogs into seven different groups. Each of the clubs may have different names for the groups, and some of them may disagree as to which group to put a specific breed in, but the basic ideas are all the same. Below are the major categorizations that nearly every purebred dog breed fits into somewhere:

All the categories below have been noded and include all the breeds associated with the category. The categories names are those used by the American Kennel Club (AKC), but I have provided the names used by other kennel clubs in parenthesis.
Herding Dog (aka - Pastoral Dog)

As the name suggests, these dogs are used for herding. This group has the desire and ability to control the movements of other animals - quite often sheep or cattle. Some of them work by stalking or staring, some by barking, and some by nipping. Some work by gathering or grouping the animals and others by moving and driving them. Most of the dogs in this group are easy to train; do their work both on their own, and through verbal commands; and are intelligent and devoted.

Hound (aka - Hound Dog)

These dogs are raised to find and chase, then catch or kill their quarry (usually small mammals). They are typically quite fast, and can be generally categorized as sighthounds or scenthounds who obviously hunt through sight or smell respectively. Unlike the sporting group, these dogs would rather catch the animal alive rather than let the human shoot it.

Non-Sporting Dog (aka - Utility Dog)

This group is typically the catch-all group for breeds that tend to defy categorization, or whose function is no longer useful. Many of these dogs have short breed histories because they were bred for companionship rather than to serve a specific purpose.

Sporting Dog (aka - Gun Dog, Gundog)

This group was bred to be a hunting companion. Its purpose is not to catch prey, but to find it, scare it out of hiding, or retrieve it once the human hunter kills it. These dogs are the pointers, setters, and retrievers.

Terrier

Terriers were bred to find and kill rodents and other vermin. Usually small, these dogs are great diggers (terra means earth you know), and may even pursue their prey underground. These dogs can be grouped into long-legged and short-legged breeds.

Toy dog

It is hard to provide any general qualities to these dogs (other than that they are small) because many of them are simply dogs from other groups that were bred simply to produce a smaller size over the generations.

Working Dog

As the name suggests these dogs are bred to do a lot of physical jobs, and are therefore typically quite large. These breeds may be guard dogs, sled dogs, cart dogs, rescue dogs, or just used for general protection. These types of dogs are typically intelligent and hardy.

If you are interested in cats instead, see Cat breeds.