Dear Paula,
                 Welcome to the forum !  That part is simply a piece of noise-reducing material which would have been glued to the metal casing to make the machine quieter when you are typing.  I'm not totally sure where it came from but my guess is either the front or back of the machine (look on the inside of the casing for tell-tale signs of glue stains roughly the shape of the part)  If you cannot glue it back in place without major dismantling of the outer casing, do not be too concerned.  The machine will work fine without it and I doubt if you could tell the difference in sound with it in place in any case !  Regarding the part that says 'Champion' (it is called the 'Paper Table') I cannot really advise without more photos/details.  The golden rule is never force anything on a typewriter !