I am a mom and flight attendant...no you do not have to have a car seat for a 2 year old, I always did for mine since they were comfortable sitting in one and knew they couldn't get out. Often when sitting in a "normal" seat small children will unbuckle and want to roam about. This could be dangerous...look what just happened with Air Canada and clear air turbulence. Sure on long flights we all need to get up and move a little but we need to be seated and belted as much as possible.
Here are some guidelines: if you take your car seat read the label...it must say for use on aircraft...some car/booster seats are only for use in cars not airplanes too. When you reserve your seats the car seat must be placed in the window seat...it can not be between any person and the aisle way. If it is a two aisle airplane (i.e. 777, 767, etc...) it may be placed in any seat in the middle section between the two aisles. The car seat may also not be in the exit row or the row in front or behind a window exit.
If you don't want to take the car seat with you...they are big, bulky and a pain to carry around, there is a new device you can purchase for use on airplanes only, not cars, it is a Aviation Child Safety Device (ACSD). They are pretty pricey, 75.00 when I looked into them, but are much easier to travel with, especially if you don't need a car seat at your destination.
Also, be sure a pack plenty of snacks, diapers, etc...you would be shocked at the number of parents I have had who don't pack anything for the kids. Food is almost nonexistant on planes now. Often we don't even have pretzels or nuts. Be sure and check with the TSA website on what you can bring through security, they do still have restrictions on liquids.