Please note that your car seat must have a legible "FAA approved" label. While I did fly AA at the time my kids were babies (last car seat used about two years ago) this IS FAA regulation and some crews want to see the sticker on the seat.

Also note that if flying internationally, rules with partner airlines, especially in Great Britain, may vary wildly. For example: BA does (did?) not allow the use of an FAA approved car seat during take-off and landing, but insisted on the baby being belted in front of an adult passenger (a sure way to kill a child in an accident). We avoided BA like the plague, as long as our kids needed the car seats in flight.

The simple rule to seat placement, as stated before, is that no passenger's egress may be hindred. That means in case of rear facing seats: window seat (all aircraft) or center seat (widebody) only. BTW: our rear-facing Century car seat did hardly fit in some economy seats at all (eg: Lufthansa or BA intra-european), blocking seatback recline for the seat in front of it.

One more tip: many baby ears react strongly to the pressure changes on bord an aircraft. Make sure that he/she has a bottle to suckle on during take-off and landing. The more the baby swallows during that time, the less the pressue change is likely to cause ear pain.

Extra tip if only one parent is traveling with a child internationally: in many caeses, a letter of consent of the other parent is required. Also, if your last name is different from the one of the child, carry a copy of the birth certificate (or any other document identifying you as parent/custodian).

Happy traveling. BTW: my son was born in 2000 and made Platinum on AA in 2001!