When I was a child, I recall an aircraft landing at my local airport, CSG, I would like to know if airports keep records of what aircraft land there? If so how long do they keep these records, and are the available to the public?
Simple answer... No. You could rule out the types of planes that could land at an airport by an airport's 5010 but there is nothing specific.
Speaking for the US, airports only report total movements (landings/takeoffs) and this information is available to the public via the FAA. This information is in aggregate and does not contain the type of aircraft.
Flightaware tracks only the type of aircrafts that land and takeoff at an airport; it also tracks the identifier used. It has very good coverage globally, although some areas have better (more accurate) information than others.
For most GA flights, the identifier is sometimes also the tail number, which can then be used to narrow down on a specific airframe.
Just add the ICAO identifier for your aerodome at the end of this URL:
http://flightaware.com/live/airport/ to see stats of a particular aiport; for your airport its http://flightaware.com/live/airport/KCSG