Consign either means to deliver something to someone (or arrange for that to be done) or to put a person or object in a place in order to be rid of them. Eg the editor consigned the manuscript to the return envelope.
A consignment is a group of items which are to be, or have been, consigned.
Con sign is a cheating or misleading sign which doesn't live up to what's written on it. Eg those which say 'up to half price sale' when in reality it's the discount which will be up to 50%, not the sale price.