What is airport tax used for?
As jwenting pointed out, there isn't one "airport tax" - there's a whole bunch of them, and they're not all taxes (some are "fees").

airlines.org has a list of the common ones you'll pay in the USA, and approximately what they cost you.  



Broadly the taxes you're paying on airline tickets break down into two categories:

Taxes that fund the Airport and Airway Trust Fund
These taxes pay for Air Traffic services (a little under half the operating butdget for ATC), maintenance of the airway infrastructure (radio navigational aids, RADAR sites, etc.), research and development to improve the airway infrastructure (e.g. NextGen), and to fund the Airport Improvement Program (which provides grants to airports, mainly to implement safety improvements).

The actual tax categories themselves break down further (again, airlines.org's breakdown is excellent).



Fees that fund DHS programs
These are often called "security taxes", and include things like the "September 11th fee", APHIS fees (for inspecting plants and livestock), and customs/immigration charges.
Note that all of these are technically fees, not taxes -- these are only levied when a flight is using the services the fee is funding (or at least that's what's supposed to happen. In practice the airline is responsible for paying the appropriate fee as a cost of doing business. They can recoup that cost any way they'd like (typically they lump it in with "taxes" or itemize it out on the ticket rather than including it in the base fare, so they can advertise lower base fares).



What about the little planes?
General aviation doesn't pay "ticket taxes", but that doesn't mean they get to escape paying for the infrastructure the flights are using. The government recognizes that the air traffic system costs money, and they like to ensure that the money is available.
Accordingly there is a tax levied on all aviation fuel purchased in the US: 


Airlines currently pay about $0.04/gallon in taxes for the jet fuel they burn  
Non-Airline flights burning jet fuel pay about $0.22/gallon in taxes
(the higher rate making up for the other taxes they're not paying)
Piston-powered planes burning aviation gasoline pay $0.19/gallon in taxes
(higher than what the airlines pay for jet fuel, but lower than the standard jet fuel tax because piston planes don't travel as far, and are thus less of a burden on the air transport system)