Besides just being one of the greatest unused band names
of all time, the ultraviolet catastrophe
was one of the great unsolved problems
in late 19th century physics.
Now, in the 19th century, most physicists thought that the science of physics was just about completed - they would have discovered everything there was to discover. (See hubris.) At the time, many physicists were simply striving for refinements to existing equations, trying to squeeze out that one extra digit of accuracy. But there were a few holes in scientific knowledge - the photoelectric effect, for one, and more importantly, the ultraviolet catastrophe.
Quick terminology for you - a blackbody is an idealized object that absorbs and emits electromagnetic radiation on all frequencies. Most equations bank off of it; otherwise, it's too hard to get down the exact characteristics of how a material reacts to radiation into a formula. Might as well stick with an imaginary something that'll approximate any object, right? Anyway. Two guys named Rayleigh and Jeans came up with two separate equations, based purely on the existing laws of thermodynamics, that generally explained how much the blackbody would radiate based on the temperature of the body and the wavelength of the radiation being measured.
Now, this should have worked beautifully - the laws of thermodynamics were tested, well known... hell, they got up to the level of 'law' in science, which takes reams and reams of evidence. To put this in perspective, evolution is still a 'theory', not yet a 'law'. And the equations worked for most wavelengths of radiation. But, as the wavelengths got small and approached zero, the energy radiated would approach infinity... and that's clearly wrong. There was a nice curve, built from empirical data, that showed how much energy would be emitted at short wavelengths, and let me tell you, it wasn't anywhere close to infinity. This was a big freak-out issue for many physicists of the time - I mean, you're almost done finishing off a branch of knowledge, and there's still this one unsolvable gaping hole? One that seems to say that the existing laws were a bit off? That's why it was called the ultraviolet catastrophe - it seemed to suggest that a lot of accepted fact was horribly, horribly wrong.
So in walks this guy, Max Planck. Contrary to popular belief, Max was pretty much conservative, not one to shake things up, ever. He wasn't seeking to cause a big fuss. But he did want to solve the ultraviolet catastrophe. So he took that empirical data (for all wavelengths) and fit an equation to it. This equation described, damn near perfectly, the amount of radiation a blackbody should put out. And most of the constants in it could be easily described (this number is temperature, that number is volume, etc.) but there was this one constant that Planck couldn't understand.
But somewhere, after a few years of staring at this equation, he realized that this mystery number seemed to suggest that the energy was being emitted in 'packets' - little discrete bundles of energy. This was completely at odds with accepted laws of physics, which held that energy was emitted continuously in waves. If energy was emitted in packets, that would seem to say that energy was carried by particles, and that was (mostly) disproved years ago...
Planck was reluctant to release this information to the world - he was fairly confident in his answer, but he wasn't sure if his interpretation of it was all that great.... but he published his results anyway. And the academic world went mad. I mean, what's with this guy, Planck, taking on long accepted laws of physics? That's like proving Newton wrong. That's saying that there's a whole load of work to be done, checking laws, seeing which ones were still true if, in fact, energy was emitted in little packets. And Planck was starting to say just that. He kicked up a really big fuss, to his embarrasment. And he did this just after the turn of the century - 1900.
The battle over this raged on for a few years, with Planck mostly getting laughed at in private and public, until some wild-haired dropout by the name of Albert Einstein happened to use Planck's constant of energy in another mystery - the photoelectric effect. The photoelectric effect is where, according to known laws, the strength of radiation should be the only factor affecting the amount of electricity it would excite in metal. But instead, it depended on frequency. Einstein, after some careful thought and some thoughtless scribbles, came up with an answer - the packets of energy hitting the metal every X seconds (X being very low, here - like, 0.000000001 or something) would do this, since X here is frequency by another name. Einstein and Planck's work buttressed each other; they both obtained the same constant for the strength of each packet. Einstein called these packets photons; they would later make appearances in Star Trek torpedoes, and elsewhere.
And slowly, these new theories (called quantum mechanics), along with Bohr's newfangled model of the atom, gained enough credence to take over the academic world. After all, these were heady times; Einstein later came out with the theories of Relativity; Heisenberg explained light's wave-particle duality (kind of) and freaked out people with his Uncertainty Principle; etc. etc. ad infinitum. The next half-century, starting from the publishing of Planck's work at 1900, was the most productive era in physics (and possibly all academics), ever. And it came after a time when most people thought that there was soon to be little work as a physicist. (Again, see hubris.)