They are the recipe for one piece of the virus (which is important to its function, so hopefully among the parts least subject to mutation), and then your cells

They are the recipe for one piece of the virus (which is important to its function, so hopefully among the parts least subject to mutation), and then your cells that get vaccine in them make the foreign protein, which then causes the usual immune response to foreign proteins. The difference from what you said is that the code in the vaccine is for the foreign stuff, and the immune stuff comes from your immune system.