The system was first introduced by the RAF who used fairly large solid fuel rockets to shoot planes (typically the Hawker Hurricane) off a small ramp. These were fitted to the fronts of merchant ships in order to provide some cover against German spotter planes. After firing, the rocket was released from the back of the plane to fall into the water (and sink). The pilot would later parachute from the plane, hopefully to be picked up by one of the escort vessels.
The Luftwaffe also used the technique in order to help their small bombers into the air with loads that would have made the takeoff run too long otherwise. Their system typically used HWK ("Walter") rocket engines burning hydrogen peroxide. A parachute at the front of the motor was used to slow its fall after being released from the plane, so the system could be re-used.
After World War II JATO became particularly common owing to the low slow-speed thrust of then-current jet engines. As the quality and power of the engines has grown, JATO has fallen from favour. It is still used, however, when heavily-laden aircraft need to take off from short runways.
In all of these cases the term "jet" is inaccurate and the system is more accurately called RATO. However JATO remains the most popular version, apparently due to its US origin.
The JATO Rocket Car is a famous urban legend that relates the story of a car equipped with JATO units for a lark, that is later found smashed into a mountainside. This story is often given as an example of a Darwin Award; however it appears to be apocryphal, with no basis in fact. A particularly elaborate form of this legend has been promulgated by hacker group Cult of the Dead Cow in the ostensibly autobiographical story "Rocket Car". This legend was convincingly debunked in 2003 on the Discovery Channel show "MythBusters". They replicated the scene and the thrust of the JATO with some commercially-available amateur rocket motors. The car did go very fast, maybe 150 MPH, but did not go anywhere near 300 MPH, and did not become airborne.