The concept of linear relationship suggests that two quantities are proportional to each other: doubling one causes the other to double as well.
Linear relationships are often the first approximation used to describe any relationship, even though there is no unique way to define what a linear relationship is in terms of the underlying nature of the quantities. For example, a linear relationship between the height and weight of a person is different than a linear relationship between the volume and weight of a person. The second relationship makes more sense, but both are linear relationships, and they are, of course, incompatible with each other. Medications, especially for children, are often prescribed in proportion to weight. This is an example of a linear relationship.
Nonlinear relationships, in general, are any relationship which is not linear. What is important in considering nonlinear relationships is that a wider range of possible dependencies is allowed. When there is very little information to determine what the relationship is, assuming a linear relationship is simplest and thus, by Occam's razor, is a reasonable starting point. However, additional information generally reveals the need to use a nonlinear relationship.
Many of the possible nonlinear relationships are still monotonic. This means that they always increase or decrease but not both. Monotonic changes may be smooth or they may be abrupt. For example, a drug may be ineffective up until a certain threshold and then become effective. However, nonlinear relationships can also be non-monotonic. For example, a drug may become progressively more helpful over a certain range, but then may become harmful. Thus the degree of help increases and decreases and this is a non-monotonic, as well as a nonlinear, relationship.
Even when a relationship is monotonic, and the changes in one quantity are smoothly related to the changes in the other quantity a linear relationship is not always the best approximation. It is often useful to generalize to a power law relationship. In a power law relationship every time you double one quantity the other one is multiplied by a number which is not two, but it is always the same number. The dependencies of quantities in many complex systems have been found to be better approximated by power laws than by linear relationships. A power law is a more general form of relationship and for this reason alone it should be a better approximation. However, in many cases there are fundamental reasons for power law behavior in complex systems.
Related concepts: Nonlinear dynamics, power law
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