If the paying customer – that person shelling out $5,000 to $20,000 per hour - cancels his or her trip, then your empty leg will also be cancelled. If you were going somewhere, make sure your hotel and car rental reservations can be changed without penalty. If you were using an empty leg to get home, make sure you also scan the airline schedules and availability so you have a backup plan. Most empty legs can be cancelled anytime before departure, and reasons can include that delays earlier in the day meant your pilots ran out of time to something on the inbound flight broke and needs to be fixed. Oh, and that’s it. You’ll get a refund of your money, but you’re usually on your own, perhaps sitting in a private jet terminal at an airport with no commercial flights.
Charter, also called air taxi or ad-hoc flights require certification from the associated country's regulating body such as the FAA in the U.S. The regulations are differentiated from typical commercial/passenger service by offering a non-scheduled service. In the U.S. these flights are regulated under FAA Part 135. There are some cases where a charter operator can sell scheduled flights, but only in limited quantities.
Of course, the cheapest way to fly privately is to have a friend who invites you on their flight. Then there are the semi-private charters like JetSmarter, JetSuiteX and JetClass where you can buy a seat on a scheduled private jet and corporate shuttle flight sharing it with strangers, much like if you were on a commercial airline. You can buy tickets on these flights for under $200 and you do avoid those crowded main airport terminals. However, if you want to have the real deal experience, chartering your own private jet, you are looking at something between $5,000 and up to $20,000 per hour depending on the size of aircraft and routing.