Nonlinear regression fits a mathematical model to your data.

What is a model? A mathematical model is a simple description of a physical, chemical or biological state or process. Using a model can help you think about chemical and physiological processes or mechanisms, enabling you to design better experiments and make sense of the results.

"A mathematical model is neither a hypothesis nor a theory. Unlike scientific hypotheses, a model is not verifiable directly by an experiment. For all models are both true and false.... The validation of a model is not that it is "true" but that it generates good testable hypotheses relevant to important problems. " (R. Levins, Am. Scientist 54:421-31, 1966)

A simple model relates two variables with a straight line. Y equals a slope times X plus an intercept. You can fit this model to your data using linear regression, to determine the best-fit values of the slope and intercept. See Linear regression. Linear regression is special, because the math is so simple and you can compute the best-fit values of slope and intercept by hand if you wanted to. Other models require more difficult calculations, but the idea is the same. When you fit a model to your data, you obtain best-fit values that you can interpret in the context of the model. For  example, you can determine rate constants, equilibrium binding constants, etc.

In most circumstances, you'll be able to use standard models developed by others. You will only need to develop new models if you work with new experimental systems, or need to extend conventional models to new situations. See Classic equations commonly used by biologists and How models are derived.