The kind with the harness that can also be used as a booster is normally referred to as a combination seat in car seat circles (also note in the link I provided that they call it a "combination booster" in the product description). They look something like this:
http://www.amazon.com/Evenflo-Chase-Booster-Seat-Optical/dp/B00AJSIXWK/ This type of seat generally can be used on an airplane in harness mode. Also, they often have a terrible belt fit when the harness is removed, rendering them useless as actual boosters (I know, I owned 4 of them between my two girls and none of them provided proper belt fit when used as a booster, so I had to buy dedicated boosters when they were too big for the harness). We actually do still own one of these combination seats, a Britax Frontier (
http://www.amazon.com/Britax-Frontier-Booster-Seat-Onyx/dp/B00BEVF1W6/) that my youngest is harnessed in for use in our primary vehicle. I do not consider her to be in a "booster" when she is in it because she is still using the harness. She is in a booster when she uses her Graco Turbobooster.
A dedicated booster, on the other hand, does not have a harness and can never, ever be used with a lap belt only. Therefore, they can never be used on an aircraft:
http://www.amazon.com/Evenflo-High-Back-Booster-Wyder/dp/B006PB2C5Y/ This is the type of seat that most people generally are referring to when they say "booster". The one in the picture happens to be a high-back booster, but backless boosters are also referred to as boosters.
The big takeaway here is that only harnessed seats can be used on airplane. A booster that relies on the vehicle belt cannot be used with a lap belt only, and therefore cannot be used on an airplane.