When you look at the heating element in your oven, you know that hot objects give off light. In fact everything at any temperature emits light. We don't usually associate this with everyday objects because the light coming from objects at room temperature is far into the infrared where we can't see it, but it is there. Another observation is that a shiny object seems to give off the same distribution of light as a dull object gives off at a lower temperature. An object which gives off a certain distribution of light at the lowest temperature possible is called a blackbody. At a given temperature, all objects give off a "redder" light distribution than a blackbody at the same temperature, and so appear to be "cooler". A large, heated cavity with a tiny hole in the side is often presented as being an excellent approximation to such an ideal blackbody. The spectral distribution is measured by "looking into" the box through the small hole (or more correctly, by allowing the light inside to escape out through the hole and then measuring it. The hole needs to be small compared to the size of the box so that the amount of energy escaping is negligible compared to all the energy inside the box. Then the measurement doesn't disturb the distribution.)
Lord Rayleigh first attempted to describe this distribution. He modeled the blackbody by assuming that it consisted of a collection of oscillators that could absorb and emit electromagnetic radiation at any frequency. By statistical processes he arrived at a certain equation, which when he presented it publicly, an audience member by the name of Jeans stood and said "It seems to me that Lord Rayleigh had introduced an unnecessary factor 8 by counting negative as well as positive integers." With that small correction came the celebrated (but still very wrong) Rayleigh-Jeans Law. The main problem with this is that though it fit the data very well at low energies (the distribution of light in the infrared region), it predicted a growth without bound as one went towards the ultraviolet end of the spectrum. This was called the "Ultraviolet Catastrophe" since it predicted that the world should be filled with X-rays and Gamma Rays and we all should have been fried eons ago.
Max Planck played around with the equations and came across a mathematical form that exactly predicted the observed blackbody spectra. However, as he tried to understand what it meant, his only conclusion was that the oscillators making up the system could not absorb radiation of any frequency. Instead they had to absorb and emit it in only discrete quantities - that the energy of the oscillators had to be quantized. This was the birth of the quantum. No explanation of WHY they were quantized was given but it certainly predicted exactly the observed blackbody emission spectra. Others undertook to explore this concept, though Planck himself never believed it was physically real. He only thought that he had developed a mathematical trick that worked. But he still gave birth to the quantum.