A potential well is typically represented in graphical or mathematical form. It is defined as the equilibrium separation distance between atoms. This is very similar to a gravity well or gravitational well. Instead of dealing with space-time, it just deals with electrostatic charge. Note that it can also apply to other repulsion-attraction systems.

Potential wells are used in analysis of ionic bonds. They result from the balance of the repulsive forces of two atoms' electron clouds upon each other, and also from the attractive forces of the net charges.

The attractive forces between the atoms decline exponentially with distance. The Repulsive forces also decline with distance, but follow a different function and are opposite in magnitude. The well is called a potential well because the potential energy between the two atoms varies as the interatomic separation is varied. The atoms will naturally settle at the lowest-energy point on the curve, that is the radius at which the lowest-energy state is reached. This is only completely true in a material at absolute zero, which is impossible. At any higher temperature, the radius will fluctuate around the miniumum potential.

The usual analogy used for this is that given a sort of concave up bit of land, (that is, a bowl shape), and a freely rolling ball. This is a decent analogy since in both cases what will move -- the ball or, equivalently, the radius -- is affected by potential energy. Started at any point on the valley, the ball will eventually settle at the bottom. (Only in the presence of friction, without which it would keep rolling up and down the sides, oscillating.) Atoms in an ionic lattice do this same thing, although they have other forces on them from the surrounding atoms. If two atoms of opposite charge (alone) are placed at a certain radius and allowed to interact, they will eventually settle at the lowest-potential radius, releasing energy along the way.

This writeup blatantly ignores several key points relating to thermodynamics, and instead uses hand waving.

This information was derived from my Materials Science and Engineering lecture notes. Special thanks go to tdent for critiques and assistance.