On a force-extension graph, a linear relationship is a straight line through the origin. The extension is directly proportional to the force, so doubling the force doubles the extension. A spring gives this kind of graph provided it is not stretched past its limit of proportionality. For example, extensions of 2 cm, 4 cm and 6 cm for forces of 1 N, 2 N and 3 N are linear, because the extension increases by the same amount for each extra newton.
In a non-linear relationship the graph is a curve. The extension is not proportional to the force, so doubling the force does not double the extension. A spring that is stretched beyond its limit of proportionality becomes non-linear, because the line starts to bend and the graph becomes a curve. An elastic band is a non-linear system even at small forces.
The shape of the line tells you how the object behaves. A straight line means the object stretches steadily with force. A curve means the extension changes by a different amount for each extra newton.
