A rubber sheet can be used to (crudely) illustrate two effects of gravity - spacetime stretching and gravitational attraction.
Take a light but strong sheet of rubber, of uniform thickness, and stretch it tight across a horizontal frame (but not so tight it has no more elasticity). This rubber sheet represents a two-dimensional universe. Place a heavy ball in the centre of the sheet, to represent a massive object existing within the two-dimensional universe. It will deform the rubber sheet, make a dent in it. Parts of the rubber sheet nearer the ball will be displaced more than parts of the rubber sheet farther from the ball. This displacement can be thought of as a physical manifestation of potential energy within the two-dimensional universe, and it corresponds exactly with potential energy in this three-dimensional universe in which we are performing the demonstration. If a small marble were to be placed far from the the ball, it would be higher than a similar marble placed near the ball. The marble farther away has more potential energy than the nearer marble. If you were to let the two marbles go, the one that starts higher will hit the ball with a greater impact than the one that starts lower.
Now place a marble on the rubber sheet and just let it go. The marble will roll towards the more massive object. That's plain old gravitational attraction working in it's most obvious form. Place the marble on the sheet, between the ball and the frame, and roll it sideways. By changing the speed at which you launch the ball, and the position from which you launch, you can "land" the marble on the side of the world opposite the launch point, you can achieve orbits of various eccentricities, and you can achieve flybys (gravitational slingshot) where the marble will hit the frame but it's trajectory has been altered so it's travelling in a different direction than when it started.
There are several shortcomings with this demonstration. The frame from which the rubber sheet is suspended, for example - there is no counterpart to that in the real universe. The frame causes objects to roll towards the centre of the sheet, and there is no centre of the universe towards which everything is attracted. The balls are three-dimensional objects, whereas for a more proper model the massive object would be contained enitrely within the rubber sheet. I don't know if the rubber-sheet model actually emulates the inverse square law accurately (indeed, it would be inverse-linear in a true 2-D universe). Most imporantly - the balls only make an impression on the sheet because they are being pulled down by Earth's gravity. 3-D gravity must exist in order for this model of 2-D gravity to work... so this model, while demonstrating parallels to some of the effects of gravity, doesn't really say anything about the mechanism of gravity. (There may be other flaws too; the setup described here is more of an illustration than a demonstration.)