The expression "fighting like cats and dogs" arose for good reason. The two species are fundamentally different and will view each other as antagonists unless steps are taken to re-engineer this perception. Though both species are highly territorial and predatory, dogs are larger and will chase cats off or even kill them under unfortunate circumstances.
Fortunately, dogs and cats do not necessarily have to be enemies, if they are properly socialized at an early age. Dogs that have not been socialized toward cats from an early age are not likely to be friendly or accepting of cats that live under the same roof. The same is also true for cats, who may love or hate dogs depending on their early experiences with them. For dogs and cats that have not been socialized together, the best that can be hoped is often a brittle peace maintained by avoidance and disparaging gestures.
In dogs, the sensitive period of learning regarding social acceptability and manners occurs between 3 and 12 weeks of age and in cats between 2 and 7 weeks of age. During this time, you can engineer many relationships that would be doomed later in your pets' lives.
It is usually not possible to raise kittens and puppies simultaneously to create this "bon accord," but a huge step in the right direction involves introducing puppies and kittens to friendly members of the opposite species during the sensitive period.
What You Can Do
If you are unlucky enough to own a dog and cat that have not been socialized, you may have an ongoing battle on your hands (as the Clintons did in the White House). One solution is to find another home for one or other animal but if you have time and patience you can sometimes make the best out of a bad situation.
Introduce the feuding pair to each other at a distance, or across closed doors, with each animal (especially the dog) under good control. While each animal is calm and relaxed, this state of affairs should be rewarded handsomely. If this process is repeated frequently, each pet learns that, in the presence of the other one, good things happen. The distance between the two should then be incrementally decreased and the time the pets spend together increased, until they can coexist peacefully.
Managing such a reintroduction program takes strong leadership on the owner's part, plus patience and good physical control. The feuding parties may never become the best of friends but can at least be taught to tolerate each other's presence.
Just in case your newly trained animals suffer a setback, it is prudent to provide a place in every room of the house to which your cat can retreat, should this prove necessary. A climbing frame, a springboard to a tall piece of furniture, or narrowly open, firmly secured cupboard door will provide safe harbor for the cat in times of emergency or exasperation. Sometimes the dog is on the receiving end, but in such instances, it will usually learn to give the cat a wide berth, thus circumventing trouble.
For peace of mind, be advised that cats don't kill dogs and the worst injury that the dog is likely to experience is a scratched cornea. Soft Paws® (plastic nail caps) or simply regular nail trims (plus nail filing) for your cat will help reduce this unwelcome complication.