Since the electron is negative in charge and the proton is positive incharge, they are attracted to one another. Thus, electrons like to orbit around the nucleus. Then, both these protons and neutrons together make up the weight of the atom, but the electron is so miniscule that it is almost weightless.
But actually, electrons weigh about 0.00054 amu. This is so much less than the mass of a proton or a neutron that the apparent mass of the electrons is ignored when the mass of the atom is determined. Thus, it is considered weightless but it does have weight that is too negligible.
They say that the electron is weightless because it is so small that it can barely be seen. There are also other reports stating that perhaps it is not true that electrons exist because there are not enough details that will show that they are actually there.
Electrons have not been photographed and people do not have the right tools in order to magnify atoms to hopefully show the electrons, protons, and neutrons. Yet, they are generally weightless because they are usually attracted to protons and they may go around the nucleus. When this occurs, the only weight that they will have is their combined atomic weight.