Just to add a bit of history: Planck started the entire "quantum business". He was very interested in the most fundamental laws. One of the open questions at this time (end of 19th century) was the spectrum of "black bodies". It was very clear from thermodynamics, the main topic of Planck's interest and expertise, that this spectrum is independent of the specific material a black body is made up (a very good "black body" is realized by a cavity with a very little hole, through which the radiation emitted from the walls, which are held at a certain temperature). It thus only depends on fundamental natural constant.
The trouble was that classical electromagnetism (Maxwell's theory) together with thermodynamics/statistical physics leads to an absurd result: The electromagnetic field in the cavity can be described as a superposition of harmonic oscillators (Fourier decomposition of the electromagnetic field). According to the equipartition law of classical statistics each oscillator contributes an energy ##k_{\text{B}} T##, where ##k_{\text{B}}## is Boltzmann's constant (one of the fundamental natural constants, which at the end just turns out to be a conversion factor from measuring temperatures in Kelvins and energies in Joules). Now there are infinitely many such field modes in a cavity, and thus the total radiation energy becomes infinite. This socalled "ultraviolet catastrophe" cannot be avoided within classical physics.
Despite the Planck's expertise in thermodynamics and statistical physics he was also very lucky that there was a great practical interest in defining an absolute measure for the brightness of light sources. Thus, the "Physickalisch Technische Reichsanstalt" (the metrological institute of Germany at this time) set up a research project to define such a standard or unit. The black-body radiation spectrum would be an excellent standard, because it's independent of the specific material and thus can be reproduced everywhere easily. Thus, Rubens, Kurlbaum et al started a research project to measure this black-body spectrum very accurately, and this helped Planck at the end of 1899 to guess the correct radiation law which fitted the accurate data over a wide range of frequencies.
Of course, for a theoretical physicist this cannot be satisfactory at all. There is a fundamental law of nature. So there must be theoretical explanation, and indeed Planck found one, using a very clever way to apply Boltzmann's famous entropy principle, according to which ##S=k_{\text{B}} \ln \Omega##, where ##\Omega## is the number of microscopic states realizing a macroscopic state. It turned out that Planck had to assume the electromagnetic radiation can absorbed and emitted by the walls of the cavity only in portions of ##E=h f##, where ##f## is the frequency of the light wave and ##h## a then new natural constant, now called Planck's constant. For convenience nowadays we write this law in the form ##E=\hbar \omega##, where ##\hbar=h/(2 \pi)## is the "reduced Planck constant".
Although this was an ingeneous finding, and it was honored with the Nobel prize of physics for 1918, Planck was very unhappy with this finding. For the rest of his long life he tried to find a way to derive his law from classical physics, but to no success.
Then a pretty busy period started, and the quantum was everywhere all of a sudden, and it solved a lot of problems (also closely related to thermodynamics but also to the upcoming atomic and nuclear physics) like the problem of the specific heat of solids, the discrete spectra of atoms, the structure of the chemical elements described by the periodic table, etc. However, the state was very unpleasant to all physicists until 1925, when a young physicist, Werner Heisenberg, found a formulation of "modern quantum theory" (then worked out by Born, Jordan, and Heisenberg himself; this we now call "matrix mechanics"). A bit later, in 1926, E. Schrödinger, working out an ingeneous idea workded by L. de Broglie as his PhD work) found another version of modern quantum theory in terms of partial differential equations, now called wave mechanics. Schrödinger could soon show that the apparently different theories by the Born group and himself was in fact the same theory. This was shown also by P.A.M. Dirac, who finally found the general formulation, which allows to use matrix mechanics or wave mechanics whenever the one or the other is more convenient for a given problem. The mathematical foundations of non-relativistic quantum theory was worked out by J. von Neumann (Hilbert-space formalism, theory of unbound operators etc.).