The key to successful cat-dog introductions is to expose them to one another gradually under controlled conditions. You want to avoid creating situations where the cat runs away and the dog's prey-chase instinct is activated. If your dog has previously lived with a cat, and your new cat has previously had positive experiences with dogs, they may progress quickly to tolerating one another. However, if you have an adult dog that has never been socialized to cats, the introduction should be a very gradual process lasting up to 30 days. In either case, train your dog to sit and stay reliably before bringing your new cat home. This may give you greater control once the introductions have been made. Remember that these steps are progressive, so go on to the next step only when you feel your dog and cat have "mastered" the previous one.
On day one, confine your new cat to his or her own room at first. After a few hours, confine the dog in a fenced-in yard, basement or separate room, and allow the cat to explore the rest of the house. Then put the cat back in his or her own room, so the dog has an opportunity to become familiar with the cat's scent. Put a baby gate up but leave the door closed.
On day two, open the door to the cat's room a couple of inches and allow the dog to sniff and see through the opening for 30 seconds. Reward the dog for appropriate behavior. Repeat this step a few times during the day. Continue to give the cat the opportunity to explore the house when the dog is securely confined out of sight.
On day three and subsequently, increase the "viewing intervals" by short increments until the dog can watch the cat quietly for a few minutes. Reward good behavior. Once this has gone well, allow the dog to view the cat with the door completely open, with the baby gate still in place, for a few minutes at a time. If the dog is tolerating the cat, go into another room. Call the dog to you and play a game with him or her. Then ignore both animals (but keep attuned to them!) and engage in some other activity. The dog will start to lose interest in the cat.
Eventually, leave the door to the cat's room open, with the baby gate still up, whenever you are at home. Always close the door when you are not present! Some pet owners will always need to keep the dog and cat separated when they aren't around to supervise, but others will find that after a couple months' probation, the dog and cat are OK together by themselves. It's far better to err on the side of caution, however, to prevent tragedy. Even after your dog and cat are peacefully co-existing, make sure that the cat's food bowl and litter box are out of the dog's reach. Keep the cat from approaching the dog when the dog is eating or chewing on a bone.