First, one word about what means Quantum :
Quantum comes from quantity. Some quantity jump from level to level and not continuously. One observes that intermediates values are not possible. Simply, they do not exist. Energy is quantified.
This phenomenon is not usual. It hurts our common sense. Nevertheless, some people know very well such quantification effects. For instance, musicians when they play harmonics on a guitar. They should pluck the strings on their middle or on integer fraction of their length (1/3, 2/5, etc). The other possibilities are only producing strange sounds and nothing nice. This come from the fact that steady wave on the string must have a node at both ends of the string, where the string is fixed. By plucking the rope at a place, which creates waves without nodes at both ends, produce noise but no harmonic sound. This quantification comes from the fact that the medium (the string fixed at both ends) reduce the amount of possible oscillations to the ones with an oscillation node at both ends.
Now, it is mainly the same thing happening to an electron trapped by a proton, which form, in this way, an atom of Hydrogen. The electron is constrained to stay close to the proton due to the attractive electromagnetic force acting between them. Knowing that the electron behaves like a wave, those constrains impose to the electron to oscillate on certain modes. Each modes have a particular energy. Intermediate energies are not possible. The energy of the Hydrogen is then quantified.