Selective Breeding.  As humans domesticated the dog, they began to see that dogs could help them in a number of ways: as guards, hunters, sheepherders, workers, and just as pets.  But these different jobs required dogs with different body types and different personalities.

For example, a working dog must be large, strong and patient.  A hunting dog must be a fast runner and have good vision and hearing.  Dogs used mostly as indoor pets should be small and sociable around people.

So, starting with the basic dog, which looked a lot like a wolf, people began to select the look and personality they wanted and breed only the dogs that showed those traits.

After many years of selective breeding, there are over 100 different breeds of dog in the US alone, including Saint Bernards, rescue dogs; Shetland sheepdogs, herding dogs; Siberian huskies, sled dogs; bloodhounds, tracking dogs; and many more. Not only do different breeds of dog look and act very differently but each is suited to a particular kind of work.

Cats, on the other hand, are kept mostly as pets and have never been expected to do any specialized work, except catch mice.  Because of this, people have had little reason to select for different personalitites and body types.  This is why most breeds of cat look and act pretty much the same.