The differences between dogs and cats goes beyond the popular phrase, "Dogs come when called and cats take a message and call you back." While it is true that dogs tend to be more dependent upon their human friends, cats can be almost doglike socially, particularly if they are indoor cats. Dogs and cats can be great friends. Dogs who dislike other dogs can be fond of cats and cats that dislike cats can be fond of dogs.
If you bring a new dog into a home with a cat, I recommend having a separate area for the dog, with a crate or a baby gate, and put the cat in a different room while the dog explores the house. A few days of separation allows the new pet and existing pets to get used to things. That way they are more likely to get along. Otherwise, the dog may scare the cat. Usually, cats will tolerate the company of a dog when they feel the dog is not a threat. After a few days of separation, the door can be opened to allow the new cat out but the dog should be kept apart so the cat can explore the house in peace. A new dog should be on a leash while the cat is around to avoid having the dog pounce on or chase the cat. A leash correction or a noise when the dog approaches the cat too rambunctiously and a treat should be given when the dog approaches calmly. Usually, after several days of monitored interactions, they become comfortable in the presence of each other. Keep in mind, even if the two are getting along for days or weeks, do not leave them alone together. Separate them when you go out. The dog should never be allowed to chase a cat. It's unfair for a cat to be in constant worry. If the dog is exhibiting behavior that is more than play, such as stalking, cornering, nipping at or standing still and very focused on the cat, be careful. That’s usually more than play and could result in real harm. That the cat can easily get away is no justification for keeping them together while the dog is aggressive.
There are some dogs who just cannot live with cats and some cats will be aggressive with dogs. Fichi, the shelter mascot’s idea of interacting with a dog includes launching himself with nails full out and landing on the dog’s head. In some homes with cats and dogs living harmoniously, they can be best friends. Sometimes they just tolerate each other. In my house, the dogs and cats can be near each other until the cat is on my lap. Then, if the dog gets too close, he gets a whack. In my experience, the key to a successful introduction is taking your time. It's far easier to take a little longer than trying to undo a bad first impression. I go over this with everyone who adopts a cat from us who already has a pet at home. If they take precautions, it's always a success. Some don’t, and then complain they are having a problem. If aggression is not an issue for your pets, all it takes is a little patience for them to get along.