

All regression methods fit a model to data to find the values of the parameters that makes the model fit the best.
Simple regression fits models with one independent (X) variable. Multiple regression fits models with more than one independent variable. Prism offers multiple linear regression. In a restricted sense, it can do multiple nonlinear regression with two independent variables. See this example of enzyme inhibition data to see how this works.
See the definition of linear and nonlinear.
Polynomial models are mathematically linear, but in Prism you use the nonlinear regression analysis to fit a polynomial model.
Linear and nonlinear regression are usually run with the assumption that the residuals (vertical distance of the points from the bestfit line or curve) are sampled from Gaussian distributions. If the outcome is counts (number of objects or events), it is usually better to assume the residuals are sampled from a Poisson distribution. Prism 8 introduced Poisson regression to Prism as options in both multiple and nonlinear regression.
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 onetime 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.