Always check with your airline if you want to use your child's car seat. Generally, airlines will need to know that it conforms to the car seat standards that are used in Europe or the United States. Airlines frequently provide a list of crieteria or standards on line that you can view when booking your flight. Bear in mind that larger car seats may not fit into the aicraft seat, so if you are concerned about this speak to your airline direct. Children under the age of 2 years do not require their own aircraft seat and are instead allowed to sit on their parents lap.
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. If you are planning on using your child's car seat in the country you are flying too, do check that it conforms to the standards in that country. For example, not all European tested child car seats are approved for use in the United States, because their testing standards are different.