Linear is not defined as most scientists would guess. A model is said to be linear when the Y variable is linear with each parameter. To understand this, you need to think about the math. Hold all but one parameter constant and also hold X constant. Now vary the remaining parameter and watch how Y changes. If the change in Y is linear with the change in the parameter you vary, and this is true for all the parameters in the model, then that model is said to be linear.
By this definition, a polynomial model is linear. Let's take the example of a third order polynomial model:
Y= A + BX + CX2 + DX3
Hold A, B, D and X constant, and look at how Y will vary as you vary C. The equation now is Y= [A + BX + DX3] + C[X2 ], with the two terms in brackets constant. That graph would be a straight line. A, B and D are also linear with Y (holding everything else constant) so the model is linear. But if you graph Y vs. X for a polynomial model, you'll almost always see a curve, not a line (it depends on what values you assign to A-D. So linear describes the model, not the graph of X vs. Y.
If the model is not linear, then it is nonlinear.
Why does it matter if a model is linear? Like linear regression, it is possible to fit polynomial models without fussing with initial values and without the possibility of a false minimum. For this reason, some programs (i.e. Excel) can perform polynomial regression, but not nonlinear regression. And some programs have separate modules for fitting data with polynomial models (linear) and nonlinear models. Prism fits polynomial models using the same analysis it uses to fit nonlinear models. Polynomial equations are available within Prism's nonlinear regression analysis.
From the point of view of a scientist using Prism, the distinction between linear and nonlinear models is not very important. Choose the model that makes sense for your data. The only issue is that with nonlinear models, it is essential to provide initial estimated values for each parameter. In some cases, these choices can be critical to getting useful results. If you choose a built-in model, Prism chooses initial values for you, and these are almost always good enough to get the job done.
While the terms linear and nonlinear have standard definitions in statistics, the term curvilinear does not have a standard meaning. It generally is used to describe a curve that is smooth (no discontinuities) but the underlying mathematical model could be either linear or nonlinear.