Gravity keeps objects in orbit. It there was no gravity then the object would fly away.
Orbits are caused by the force of gravity combined with the speed of the object in the orbit. Saturn's rings consist of millions of small rocks in orbit round Saturn.
Can an object stay in orbit around Saturn somewhere else besides the rings
It depends on the altitude the object is above Saturn
Planets orbit around the Sun and moons orbit around planets. The gravity is what keeps them in their orbit. Moon of an object that orbits a planet.
Gravity combined with the object's "sideways" (tangential to its orbit) motion. The Sun "wants" to pull the object towards it and the object "wants" to fly of into space. When these two things are balanced the object is in a stable orbit.
Because of Gravity and Inertia. Inertia keeps an object moving. And Gravity keeps an object close to the main object it is rotating around.
Gravity keeps the orbiting object in orbit. If the object has more than a certain speed (depending on the distance from the central object, and the central object's mass), the "orbiting" object would escape. If it has less speed, it remains in orbit.
Gravity from the object (e.g. planet, moon e.t.c.) that they are orbiting.
Saturn's orbit shape is elliptical