Potential difference in a circuit is a collective effect of a large number (order of 10^23) electrons moving in conductors and resistors . It is a term within classical electrodynamics, and the value emerges from what happens at the level of particles and atoms, the quantum mechanical level.
This link shows what is going on at the microscopic level:
In a conductor the electrons exist in practically continuous energy levels that make them act as free, they have a drift velocity, much smaller than the velocity of light which is the velocity of the collective effect,
this shows more detail of how resistance is seen by individual electrons.
The individual electrons "see" the potential difference, i.e. the imposed electric field at the classical level and acquire their drift velocity, added to the statistical velocity of the fermi level.
Reading up on the band theory of solids will help to clarify this microscopic picture, as electrons in circuits are not tied to atoms but to the lattice as a whole.