I have tried two methods in the past and they both were very successful.
First Way (both cats were a year old):
The new cat was kept in her own room set up with her own bed, litter box, food, and toys. The other cat was allowed to roam the house since he was there first. They would sniff each other through the door. The new cat could care less about the other sniffing around.
I had a screened in porch with a sliding glass door. So I took the new kitty outside and left the other inside. I stayed out with the new cat and had a friend sit with the other. We slowly opened the door. At first just enough so they could smell each other. When they stopped growling the door was opened enough that they could stick a paw through. We used a string toy that they could both bat at and play with. Slowly the door was opened wider and wider as the cats became more comfortable. Finally all the way so they could be together.
Rarely will this work so quickly. These cats were like long lost lovers. Usually you will need to do this several times over a period of a week or so before they can be together all the time.
Second Way (2 month old kitten and 5 year old cat):
New cat was again put in its own room with its own belongings and other cat was allowed to roam the house. I would take out the new cats bedding every night so the resident cat could smell it - the new cat got the resident cats toys to play with so they could get used to each others smell. Every night I would bring the kitten out in a carrier so the other cat could see him. If it looked like she was going to go after the new cat I would take him away.
The new cat loved to play and would stick its paws under the door. I put a mouse there to get him to keep doing it. The resident cat was not the friendliest and would sit outside of the door and growl. Eventually she was more interested in playing with the mouse or the other cats paws than being angry.
As they continued to play through the door of the kittens room I would open it just enough that they could put their paws through. It took about a full week of this before all growling stopped and I finally just opened the door and had a friend with me to help me separate them if things got ugly. They didn't and started playing and rough housing (not fighting.)
For the next three days they were allowed together to play with supervision and when they would get sleepy there were separated to sleep in peace and also at night so I could sleep. Eventually they were able to sleep around each other and were allowed to stay together all of the time.
I hope this was helpful!