The goal of nonlinear regression is to fit a model to your data. The program finds the best-fit values of the parameters in the model (perhaps rate constants, affinities, receptor number, etc.) which you can interpret scientifically.
Choosing a model is a scientific decision. You should base your choice on your understanding of chemistry or physiology (or genetics, etc.). The choice should not be based solely on the shape of the graph.
Some programs (not available from GraphPad Software) automatically fit data to thousands of equations and then present you with the equation(s) that fit the data best. Using such a program is appealing because it frees you from the need to choose an equation. The problem is that the program has no understanding of the scientific context of your experiment. The equations that fit the data best are unlikely to correspond to scientifically meaningful models. You will not be able to interpret the best-fit values of the parameters, so the results are unlikely to be useful.
Letting a program choose a model for you can be useful if your goal is to simply create a smooth curve for simulations or interpolations. In these situations, you don't care about the value of the parameters or the meaning of the model. You only care that the curve fit the data well and does not wiggle too much. Avoid this approach when the goal of curve fitting is to fit the data to a model based on chemical, physical, or biological principles. Don't use a computer program as a way to avoid understanding your experimental system, or to avoid making scientific decisions.