What kind of charge does an object have if it has gained electrons

It depends on whether the object was charged before the electrons came along.

-- Electrons carry negative charge.

-- If the object was originally neutral, then the extra electrons brought extra negative charge with them, and the object became negatively charged.

-- If the object was positively charged to begin with, then it could be
charged either way after it has gained electrons.

... If there aren't enough electrons to completely cancel out the original
positive charge, then the object will still be positively charged (although
not as much) after it has gained the electrons.

... If there MORE electrons than it takes to completely cancel out the
original positive charge, then the object will be negatively charged after
it has gained the electrons.

... If there are EXACTLY the right number of electrons to completely
cancel out the original positive charge, then the object will be neutral
(no charge) after it has gained the electrons.