that's a very broad question. because all the branches of physics are intertwined with each other.
Umm.. a basic principle of quantum theory is the observation that the smallest pieces of matter, electrons, exhibit properties of both particles and waves. What an electron is doing at any given moment is a mystery to all of us. We can't just "look" at what the electron is doing to find out, though. The problem is that when we look at something, we're bombarding the electron with photons (light particles that have no mass, but carry momentum & energy). Hitting the electron with a photon changes the path, speed, etc of the electron and defeats the whole purpose of looking at it.
Instead we have to look at the end results of what electrons do - so there is an experiment called the double slit experiment which shows us where electrons hit a detector after passing through two small slits.
You might imagine that throwing electrons that passed through two holes would create two big blotches on the detector.
But the results of this experiment showed that the electrons hit the detector in a particular way - one that demonstrated that the path the electrons took was determined by wave-like motion, not like particle motion (diffraction).
Wierd - so actually, what we conclude is that the electrons took EVERY possible path from the source to the detector, and then upon observation, the resulting pattern arranged itself according to a probability distribution. Strange, yes.
Anyway, exploiting the fact that electrons' actions are governed by probability, we are able to conclude a lot of rules about electrons. Where they are, what their energy is, what their momentum is, .. all related to probability. And also all related to "standing waves." (very very complicated mathematics go here)
The thing about standing waves (imagine a violin string) is that they can vibrate at certain harmonics - either one half wave across the whole string, or one, or one and a half, as long as both ends are clamped down. But they can never be in between those. (Try it by shaking a rope faster and faster)
ANYWAY so this tells us that the amount of energy that electrons can carry can only be in certain amounts and never in between. When an electron gains or loses energy, it can only be enough to send it to the next lowest/highest energy level. Never in between. This is known as quantization, and you may know this as the "energy states" of electrons.