Dear Lumbini A Tambat:
Thank you for pointing out one of the simple concepts that get overlooked in teaching. Teaching chemistry especially! Stability. With reference to what? It's easy to imagine so many ways to get un-stable, but how can you get more stable, if you are an atom?
Well, darn it, a sodium atom is quite stable, all by itself. You can shake it, heat it, freeze it, bang it (within limits), and it remains quite unchanged. Sodium metal at 0 $^o$C and 1 atmosphere pressure is in its ground state, as stable as it can get because there is no state more stable. This is Standard Temperature and Pressure. (https://www.thoughtco.com/difference-between-standard-conditions-state-607534) Of course, we have to complicate it a bit, so we also have Normal Temperature and Pressure (NIST uses 20$^o$C), and there are 21 different modifications of the exact temperature and pressure that constitute a standard or normal temperature and pressure. https://en.wikipedia.org/wiki/Standard_conditions_for_temperature_and_pressure
But the point is that there is some generally well-defined set of conditions under which we classify materials as being in the lowest energy state possible, or most stable, or in a ground state, from which we do all our calculations and reactions.
It is possible to take two materials, both in their respective (lowest energy) ground states and allow them to react to form a new material which is now more stable because it has released some amount of energy - if you consider the two original materials, e.g., sodium and chlorine, to be in their ground states, imagine that the energy released digs a hole to an even lower ground state. That lower energy state exists because Na$^+$ ions are strongly attracted to Cl$^-$ ions in a sodium chloride lattice, and that energy of stabilization was enough to knock an electron off a sodium atom and force it onto a chlorine molecule, splitting it into pieces, one of which become a chlorine anion.
Now we explain the final situation as having a stable sodium cation, so it must have stabilized (because it can't get changed any more under ordinary chemical reactions). So now it's really stable. Unchangeable. Well, you could dissolve it in water and hydrate it, and it could become a little more stable, but not much. And the chlorine atom has an extra electron and seems quite stable: it doesn't explode or anything. But were you looking when the sodium and chlorine were reacting? They didn't look at all like they were getting stable - it would have been quite exciting!
In the picture, NaCl is the lowest energy material. The sodium is "stabilized" by the chlorine. This level is the ground state for NaCl. The next line above is a ground state for two elements, sodium and chlorine. They are stable in themselves, if kept separate, but they have reactivity, which also conveys the idea of instability. So which is it? Stable or unstable? This is where the teaching function often breaks down: one word means two entirely different things, yet we manage to communicate successfully - if we have been forewarned. We still sometimes trip ourselves up.
In the picture, we start with stable Na (solid), put in energy to vaporize it to Na (gas), put in more energy to ionize it to Na$^+$ plus an electron, then shift to the chlorine, dissociate it to atoms by putting in energy, then let a chlorine grab the electron (giving off some energy), then allowing the sodium ion to condense with the chlorine ion to form a lattice of NaCl and give off the lattice energy, which makes the final product (NaCl) so very stable the we have tons of it in the sea and in salt mines all over the earth.
Now if you have a supernova available, you could turn the sodium into iron, which is really the lowest energy you can get to, but in most labs, the diagram illustrates several levels of energy, and conditions where the materials are stable, but could react to form something more stable. Sodium is reactive and stable at the same time, depending on the conditions.