A relationship is linear if its graph is a straight line. For a spring within its limit of proportionality, the force-extension graph is a straight line through the origin, so the extension is directly proportional to the force. Equal increases in force give equal increases in extension.
A relationship is non-linear if its graph is a curved line. The extension is then not proportional to the force, and doubling the force does not double the extension. This happens when a spring is stretched beyond its limit of proportionality: each extra newton produces a larger extension, so the line bends over and becomes less steep. An elastic band does not obey Hooke's law, and its force-extension graph is a curve even for small forces.
You can tell the difference from a table of results by dividing the force by the extension for each reading. For example, 1 N for 2 cm, 2 N for 4 cm and 3 N for 6 cm all give 0.5 N/cm. Because the answer is constant, the relationship is linear. If the answer changes from reading to reading, the relationship is non-linear.
