In 1938, German chemist Otto Hahn discovered the fission of uranium. He didn't want to believe it when he discovered it, because, well, no one had ever broken up an atomic nucleus like that before. It was well-nigh inconceivable.
A quick refresher if you haven't thought about atoms in a while. Basically, they consist of protons and neutrons (together called "nucleons") packed into a tiny, tiny space called the atomic nucleus. These nucleons are surrounded by electrons--the classical picture is of them whizzing about the nucleus; that's not precisely how it works, but it's enough to know that they are separate from the nucleus and, under anything approaching normal conditions, never visit there.
Atoms are mostly empty space. A common analogy is to think of a large building--a domed sports arena, for example--if the walls of the arena are where the electrons are, the nucleus would be in the center of the building, but only the size of something pretty small, like a penny. Everything else? Nothing. Absolute nothingness.
The electrons are, of course, negatively charged, have extremely little mass, and control how atoms behave chemically--what kinds of bonds they make, what kinds of compounds they form, how chemically reactive they are, and so on. The neutrons in the nucleus add mass but have no charge. And the protons are positively charged, have very slightly less mass than neutrons, and give atoms their identities. An atom with one proton is hydrogen, one with two protons is helium, three lithium, four beryllium, and so on throughout the periodic table.
Now, you'll note that, having played with magnets, the similarly-charged ends don't like to be next to one another. It's the same thing with protons. They don't like being crammed in there together, but an extremely strong fundamental force called the nuclear force keeps them from flying apart. And that's a good thing, too, because it makes atoms, the basis of what we know of as ordinary matter, possible. Without atoms, we couldn't have coffee, or beer, or daffodils, or any of the things that make life enjoyable. Oh, we wouldn't have any kind of life we recognize, either, since we all tend to be made up of atoms.
Well, that nuclear force only acts over a very, very, VERY short distance. If you make an atomic nucleus too big--by, say, adding more protons and neutrons--it will break apart. That's really all fission is. Hahn and his assistant Fritz Strassman would shoot neutrons at atoms of uranium 235, an isotope of uranium; the element always has 92 protons, but a varying number of neutrons. Uranium 235 has the requisite 92 protons to make the atom uranium, and 143 neutrons.
Occasionally, one of the uranium nuclei would absorb an extra neutron, creating a short-lived isotope known, naturally enough, as uranium 236 (again; 92 protons, but now 144 neutrons). And then it did something amazing. It split in two.
Well, usually two. It's an entirely random process, but most fissions of uranium 235 result in two "daughter" atoms (most often isotopes of barium and krypton), three neutrons, and quite a bit of energy. If one of those three neutrons strikes another atom of uranium 235 and causes another fission, and so on in a one-to-one ratio, that is what is called a "critical" nuclear reaction. It's self-sustaining. If it's a less than one-to-one ratio, that's "subcritical," and, of course, a greater than one-to-one ratio is "supercritical."
Now, naturally, the whole process is vastly more complex than this little sketch, and you'd want to know about gamma rays and beta decays and antineutrinos and all that other good shit to get deeper into the weeds here. But I think this should suffice for a general introduction.
But back to the energy released--where does that extra energy come from? Well, if you remember your Einstein, he said that a very tiny bit of mass contains a great deal of energy. When the fissions occur, a very tiny bit of mass does get converted to energy. And we can use that energy to do things!
With a controlled critical chain reaction, you can use that energy to heat water, which you can then use to make steam, which can turn turbines and change mechanical rotational energy into, say, electrical energy. Or maybe you can use it to turn shafts that have propellers attached to them, and drive ships through the water.
Humans being humans, though, the first practical application thought of for fission wasn't a controlled chain reaction to make energy, but an uncontrolled reaction that, just seven years after the discovery of the process, was used to level a couple of cities.