What does 'Voltage is the energy per particle' expressions mean?
Isn't every particle have a certain amount of electrical energy?


No. Electrically charged particles have more or less energy depending on where they are located relative to other charged particles.

To push two electrons close together requires energy. To pull an electron away from a proton (or positively charged atomic nucleus) requires energy. 


  How can an electron have a different amount of energy.


As an analogy, think about gravitational potential energy.

A bowling ball at the bottom of a hill has less g.p.e. than one at the top of a hill. To push the ball from the bottom to the top of the hill, you will have to do work on it, that is, apply a force as it moves from one place to another. The work you do on the ball adds to its g.p.e.

If you wanted to push a tennis ball from the bottom to the top of the same hill, it would take less energy, because the tennis ball has less mass. 

With this in mind, even if there's no ball around, you can talk about the gravitational potential function of the space where the hill is. The potential is higher at the top of the hill and lower at the bottom of the hill. This lets you predict how much energy it will take to move a mass from one place to another, even though we don't know yet whether the mass we will want to move is going to be a bowling ball or a tennis ball.

Same thing in a circuit. If the potential difference between two places is 1 V, we know that if we want to move 1 C of charge from the lower potential location to the higher potential location it will require 1 J of energy. If we want to move 10 C between those two places it will take 10 J (assuming that moving all this charge around doesn't change the potential, like when we're charging a capacitor).