There is an international airport about 10 miles from my home. Is there any way I can find out online what private jets are parked there tonight? I am not familiar with any of the lingo used here :) so if you could respond in layman's terms I'd greatly appreciate it.
There is not simple list available online, apart from calling the airport and asking them directly. Aircraft are expensive assets so when people own them they tend to keep them under the radar and out of sight and out of the elements by parking them inside a hangar. As an outsider you can only see the outside parked aircraft from the airport fencing.
Determining which aircraft are parked for a short term on an airport is relatively simple to find with the use of an aircraft tracking site such as FlightAware and FR24. On these sites you can enter an airport and see all the official flights in and out of that airport. Flights that have landed but not yet departed will be parked somewhere on the airport. For example, Lubbock Executive Airpark (Lubbock, Texas, USA) https://flightaware.com/live/airport/F82/arrivals
Starting from January 2020, the FAA dictates that all airborne flights in the USA should be equipped with an ADS-B transponder, when inside ADS-B airspace, used by the sites above and air traffic control to track flights. So from that day the tracking of smaller aircraft will be greatly improved.