This depends on what elements are added during manufacture but generally the 300 series (which contains nickel) is negligibly responsive to magnetic fields. Therefore the common 304 and 316 grade stainless steels are not magnetic. When they cool after manufacture, the iron remains in the form of austenite (gamma iron), a phase of iron which is nonmagnetic. 304 grade is widely used for cookware and 316 grade for marine applications.
The 400 series stainless steel (contains chromium and no nickel) is magnetic. 430 grade is used in washing machines drums for example.