Children under the age of 2 years do not require their own aircraft seat and are instead allowed to sit on their parents lap. Using a car seat to support your child is not always possible. It depends on the car seat, and which operator you are flying with. Some airlines to not allow any car seat on board their aircraft while other are more accommodating. Some will charge. Check with your airline before flying.
Car seats that are ‘one piece’ tend to be for smaller children because of their size and shape. Of these only a handful have the certification required for use on board aircraft. The FAA have a helpful video which explains how car seats are fitted on US registered airlines. Child booster cushions are generally not allowed on board due to the dangers of the child submerging under the seat.