An empty leg flight (also known as a "dead-head" or "repositioning leg") is when a private jet flies without passengers because it's either returning to its home base after dropping off clients, or repositioning to another airport to pick up new clients for a booked charter. 

Empty leg flights can be significantly cheaper, often ranging from 30% to 75% off the cost of a full private jet charter.

Empty leg flights are almost always one-way. They are created by a specific repositioning need. You would need to book two separate empty legs if you found them for a round trip, or combine an empty leg with a standard charter.