Any charge has a certain amount of potential energy because of its proximity to another charge - this is defined by a) the amount of charge of the two particles and b) the distance between the two charges. This can be complicated by the presence of a third, fourth, etc charge, but keeping it simple by considering only two particles of fixed charge, the amount of potential energy of one charge will remain constant if its distance from the other is fixed (it may still move, but as long as the radial distance does not change, the potential energy of the charge is constant). Therefore around a fixed charge there is a shell-like range of positions that the other may have wherein it has constant potential energy. This is the equipotential of the fixed charge. In this simplified two-charge system, if the distance of one charge from the other fixed charge changes, it moves to a different equipotential and so its potential energy will change.
Now, like I said, the amount of potential energy depends on the charge of both of the particles. However, if you have one particle q1 of fixed charge and you were to place another particle q2 somewhere near it, q2 will have a certain potential energy due to q1. If you double the charge of q2, you double the potential energy of q2. Therefore you can say that at a certain distance from, or on a certain equipotential of, a fixed charge, the amount of potential energy of any other charged particle PER CHARGE of that particle is fixed. This fixed value at each equipotential is the voltage of that equipotential. So at its most basic level, voltage is the amount of potential energy PER CHARGE of a particle that the particle will have at a given distance from a fixed charge.
However, values of potential energy are arbitrarily assigned - we choose infinite distance as have zero potential energy by convention only. We could equally choose zero to be metre, a kilometre, a millimetre... whatever. However, regardless of at what distance we choose to have zero potential energy, the change in potential energy as a charge moves from one equipotential to another is the same (i.e. you can tell how much potential energy has increased ot decreased as a charge moves a certain distance without knowing what the initial or final potential energies actually were). Hence we are generally interested in changes in potential energy, and therefore differences in voltage.
