Dr. Odine explains how he created the machine.
Junction Machine Ellone is a device that exists in the distant future era in Final Fantasy VIII, though it never makes an actual appearance. It was created by Dr. Odine after intense study of a kidnapped Ellone, who possesses the unique ability to send a person's consciousness back in time into the mind of a person she knows. Dr. Odine created the Junction Machine Ellone to replicate Ellone's power, and the device proved successful though not to the extent of Ellone's natural ability. Odine had discovered a pattern from Ellone's brainwaves and found a way to mechanize it.
The Final Fantasy VIII Ultimania[1] explains that Ellone's ability to connect to the mind of another, is a same kind of process as when a person junctions a Guardian Force; one's consciousness can't be connected to another without an intermediary with a special ability—a Guardian Force, Ellone herself, or the Junction Machine Ellone.
The machine is used by the main antagonist of Final Fantasy VIII, Ultimecia, to send her mind back in time from the future and into the past where she is able to take possession of the bodies and minds of sorceresses Edea, Rinoa, and Adel, to further her goal of initiating time compression. The structure and appearance of the Junction Machine Ellone remain unknown, as does the fact of why Ultimecia seems capable of possessing only sorceresses.