Answer:
First, make sure the new cat is free of diseases. Have him vet-checked. Get all necessary shots, etc.
The cat who has already been living in the home is given priority. Put the new cat in a separate room with the door closed; they will get used to each other's scent by smelling each other under the door. After a few days to a week, hold the door open a crack and let them touch noses. Expect some growling and hissing, but be ready to close the door quickly. Do not end separation, even if they act cordially; cats can take several minutes to react negatively.
After a few days of letting them see each other through a cracked door with you there, try to create a "fence" in the doorway with a baby gate. Let them see and smell each other for 15-20 minutes at a time, with a barrier.
Once they are both acting calmer, bring the resident cat into the new cat's room -- but only with you there. You will act as protector, defender, referee, and bouncer. If they act too testy, get your resident cat back out and close the new cat back in its room. Try again another day.
It can also help during this time to switch cat bedding, so each cat smells the other cat; switch every couple days so there's a strong scent.
You can also trade litterboxes, or at least at a "present" -- i.e. use a scoop to pick up formed stool from one cat's box and put it in the other cat's box, and vice versa. Cats "greet" by butt-smelling so the litterbox droppings make a great calling card introduction. But, be sure neither cat is ill or has worms because nosing another cat's excrement is a sure way to pass illnesses between cats.
Once both cats seem to be accepting each other, start allowing the new cat into the main living space. Now your role will be to reinforce your original cat's right to be dominant, and act as protector/referee to both cats. You may need to help the new cat through its fears or insecurities too. Your goal, though, is to eventually have the two figure out the alpha-cat position on their own, to set rules together for how they behave toward each other, and what each will tolerate. To that end, you may need to allow spats, as long as no one is getting hurt.
Note, it is really important that the original cat not feel overly fearful. After all, it was his or her home, first. Help your original cat feel loved, special, and secure as if your "first born child" while helping him or her become more accepting of the other cat.
In the best intros, both cats either react like "who cares" or they love each other quickly. Unfortunately, with cats there are few best intros. Much more often, cats have to go through the growl-hiss-spat phase before they agree to be friends.