Paralyze is the American spelling of paralyse. Both mean to make a person or an animal unable to move a part of the body, or unable to move at all — the muscles will not obey. Doctors use it for a real medical state. In ordinary talk people also stretch it: fear, shock, or a huge choice can “paralyze” you, meaning you freeze and cannot act, though your body still works. The British verb is paralyse; the related noun paralysis is spelled the same on both sides of the ocean. American writers keep the -ze in the verb (paralyze, paralyzed); British writers keep -se (paralyse, paralysed). You will see both in Taiwan textbooks, which often mix US and UK spelling, so the kind thing is to treat them as one word with two coats.