Before choosing nonlinear regression, make sure you don't really need another kind of regression. Also read about how nonlinear regression differs from linear regression.
A polynomial model has this form: Y= A + BX + CX2 + DX3 ....
Like linear regression, it is possible to fit polynomial models without fussing with initial values. 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 and nonlinear regression. Prism fits polynomial models using the same analysis it uses to fit nonlinear models. Polynomial equations are available within Prism's nonlinear regression analysis.
A multiple regression model has more than one independent (X) variable. Like linear and nonlinear regression, the dependent (Y) variable is a measurement.
A logistic regression model is used when the outcome, the dependent (Y) variable, has only two possible values. Did the person get the disease or not? Did the student graduate or not? There can be one or several independent variables. These independent variables can be a variable like age or blood pressure, or have discrete values to encode which treatment each subject received.
GraphPad Prism does not perform logistic regression.
A proportional hazards model is used when the outcome is whether or not a one-time event (often death) occurred. One of the independent variables is time, and other independent variables can be used to account for treatment or other variables.
GraphPad Prism does not perform proportional hazards regression.