Most likely. When a pet travels in the cabin, the airlines call the pet "accompanied baggage." If you want your pet to travel in the cabin as accompanied baggage, you must be a passenger traveling on the same flight as your pet. Most airlines place a limit on the number of pets allowed in the cabin, so make sure you inform the airline when you make your reservation that you want to bring your pet on board with you. At that time, ask for the airline's rules on pet travel, including the recommended dimensions of your pet's carrier and the types of pets the airline will allow in the cabin.
Just because the airline allows you to travel with your pet, that doesn't mean that you can spend the flight playing fetch. Federal law prohibits you from taking your pet out of the carrier while you are in the plane's cabin.