Most airlines don't charge any additional fees for bringing a child car seat on the plane as long as you've paid for the seat on the airline for the child.
The car seat must be FAA approved and then you can install the car seat on the plane before take off.
Children that are under the age of 2 don't usually require use of a child car seat or to sit in a seat of their own and can usually ride on the parents lap in the airplane.
If they're 2 years old or older then require their own airline ticket and seat and usually must fly in a child car seat.