What is the relation between stability and lower energy? Most of the time I hear that atoms get bonded, in order to have lower energy than the separated form and that this lead them to be stable.  What makes me confused is: What exactly relates stability to having a lower energy?
 A: Think about what happens to an isolated molecule.  If a system (such as a molecule) has less energy than the dissociated constituents, then the system cannot break apart without an input of energy from another source.  A H$_{2}$ molecule (in its ground state) has less energy in its rest frame than would two free H$_{1}$ atoms.  So unless the molecule interacts with something else, receiving energy in the process, the dissociation of the two hydrogen atoms is impossible.  And that is exactly what it means to be stable:  If the system is left alone, its state will not change.
A: while Buzz has explained what it means to be "stable", I think it's a little misleading saying atoms form chemical bonds in order to become more "stable". Atoms are not conscious beings that know what will lead to more stability. chemical bonds are formed in-order to decrease the electrosatic potential energy of the system, and doing so result in a  decrease in the TOTAl MECHANICAL ENERGY of the system, which is the cause of greater "stability"
It is much better to understand this process from the point of view of physics. From physics, you may know that every conservative force (such as the electrostatic force) has associated with it a potential energy and this conservative force always points in the direction of decreasing potential energy. This is because of how the potential energy is defined i.e change in P.E=-work done by conservative force. A little mathematical manipulation can show that the conservative force is the negative derivative of potential energy.
Now as 2 atoms approach each other, there is a conservative net electrostatic force of attraction between them which pulls them together and decreases the potential energy of the system. Now, the potential energy continues to decrease as the atoms get closer together until the attrative forces between the electron densities and nuclei are just balanced by the electron-electron & nucleus-nucleus repulsions, so the net force becomes 0 and POTENTIAL ENERGY reaches a local minimum ( the stationary point in the graph)
Now, from the law of conservation of mechanical energy, the decrease in potential energy at the stable equilibrium point results in a corresponding increase in k.e, causing the system to oscillate about the equilirbium position like a spring block system in SHM. Eventually, the oscillations are damped, the excess k.e of the system is dissipated as heat, and the total mechanical energy of the system decreases.
The eventual decrease in mechanical energy is the cause of the frequently mentioned "increase in stability". Atoms form bonds of attraction because the electrostatic force pulls them together until the attractions and repulsions balance.
note: this is a crude classical explanation to build up intuition, and the actual quantum process is much more complicated.
