You are right up to a point about the charges on an atom,but you don't seem to understand what charge itself is. Charge is the source of a force vaguely similar to gravity, but trillions of times stronger. This force is called the electromagnetic force. Few people realise how powerful it is. In particles which can be observed in isolation, it is always the same as the charge on the electron or multiples thereof, and may be positive or negative. An alpha particle has a double charge because it contains two protons, which each have the same charge as an electron but of opposite sign.
To return to magnets, just as for every positively charged particle there must be a negatively charged one, for every south pole there must be a north pole. You can tell which is the north pole of your magnet by putting it against the north pole of another magnet. If it repels, you have put two north poles together, but if it attracts, you have put your south pole against the north pole of the other. It is the electromagnetic force emanating from the particles within the magnet which creates this repulsion or attraction. If it doesn't seem to be trillions of times more powerful than gravity, that's because your magnet contains equal numbers of negatively charged particles which cancel out the effect of the positive charges.