We show that any nonlinear differential-algebraic system can be locally transformed into zero dynamics form, which is a normal form with respect to the input-output behavior. Only mild assumptions on the maximal output zeroing submanifold are required and thus the zero dynamics form even generalizes the Byrnes-Isidori form for nonlinear systems with existing vector relative degree. Left-and right-invertibility of the system can be studied in terms of the solution properties of a subsystem in the zero dynamics form. This is the basis for the investigation of various classical control problems, such as output regulation and trajectory tracking.