Consider how “mass” works.
Mass isn’t a “force”, but two masses absolutely have gravitational forces that pull them together based on “how massive” each one is and “how far away” the two masses are from one another.
Consider that “charge” is very similar.
Charge isn’t a “force”, but two charges absolutely have electrostatic forces that pull them together based on “how charged” each one is and “how far away” the two charges are from one another. (Electrostatic forces are also interesting in that they will be repulsive or attractive depending upon whether the charges match or mismatch, respectively.)
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As for “What does the quantity refer to?” its *units* are in terms of electron-charges (-1) or proton-charges (+1); they have the same unit value and differ only in +/-.
An atom with +1 charge means that, if you add up all of the +1s from it’s protons, and all the -1s from it’s electrons, there is still +1 left over. (e.g. 5x +1s and 4x -1s leaves +1 overall, but 5x +1s and 2x -1s would leave +3 overall.)
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