The easiest way to think about them is that they are points that collectively make a line where if you were to move a charged particle from one point to another, it would experience no change in potential energy (and therefore require no work nor release any energy).
If you look at Figure 8-5 on page 121, the grey circles are equipotential lines. The force on a charged particle will be the same at any point on a given circle. So moving around the circle results in no change in potential energy. However, if you move from an outer circle to an inner circle, there will be a change in energy. In the system on the left, if you move from an outer ring to an inner ring, it is a natural flow, so the particle will gain KE as it moves from a higher potential to a lower potential. In the system on the right, if you move a (+) from an outer ring to an inner ring, it is a natural flow, so the particle will gain KE as it moves from a higher potential to a lower potential. In the system on the left, if you move a (+) from an outer ring to an inner ring, it requires work, so must pump energy into the particle to move it from a lower potential to a higher potential.
Hope this helps.