What keeps the electrons revolving around the nucleus of an atom?
This question goes right to the heart of why quantum mechanics became the model for describing what happens at the microscopic level of matter and energy. At the beginning of the 1900's, it became apparent that the best way to model an atom was to conceive of a very dense central part (the nucleus) with a positive charge (and almost all the mass) surrounded at a relatively large distance (compared to the size of the nucleus) by a cloud of elementary, negatively-charged electrons. These very light electrons would be in orbit around the nucleus. The problem, though, was that according to the classical physics of the time, if an electrical charge is accelerating (as anything spinning around in an orbit does), then it must radiate energy (as light). A calculation shows that in that case, the electrons would spiral into the nucleus within a tiny fraction of a second. Since the electron clouds are responsible for chemical bonding, this means that all matter would simply collapse and the universe we live in would cease to exist. As we also know, this doesn't happen. So what's going on?
In all its glory, the full answer (as we understand it) is very complicated and has lots of subtleties. Some very large-brained people worked on this problem and worked out a model that became known as quantum mechanics. It describes atomic phenomena to a precision never known before in science. It really works! However, it also has the trademark of being extremely mathematical and abstract. Still, I think I can tell you about one way about thinking about this stuff which is pretty close to the actual truth.
It is well known that musical instruments operate by the principle of resonance. That is, they are so constructed that only certain sounds can be made by them. Sounds are waves and, as such, can be mathematically described by their size (loudness) and their frequency (pitch). Musical instruments will only make certain frequencies and combinations of frequencies. That is what makes a guitar or an organ sound as they do. Now back to the electrons in the atom. It was found that the electrons could not orbit any old way they wanted to. They could only orbit at certain fixed energies (just as the guitar can only emit certain frequencies). Again, the scientists used the idea of a wave to describe this, but in this case, the wave was about the probability of where the electron could be. One could not pin down the exact position in space for the particle, but one could know exactly what energy the electron could have.
The bottom line is that the old picture of the electron spinning around in an orbit (like a tiny solar system) is simply not right. The electrons are allowed to exist at certain very precise energies, but their position is spread out, described by this "wave of probability." If you use enough outside energy to eject an electron from an atom, then this probability wave collapses to a much smaller size, thereby allowing you to know its position much better than within an atom. Then the electron tends to act a lot more like the billiard ball picture that we like to use. The usual question is - why not simply look very carefully inside the atom to "see" the electron? It turns out that in order to "see" the electron, one has to bump it (in physics lingo - change its momentum). But by bumping it, one changes its position in space. One finds that there is always a tradeoff - if you can get its position real good, then you've bumped too hard, and vice versa, one can bump very softly but then find that you cannot know its position so well. This is called the Heisenberg Uncertainty Principle (and is the source for the fictional Star Trek device known as the "Heisenberg Compensators"). Interestingly enough, there is a General Uncertainty Principle which actually defines, for a particular physical system, what things can and cannot be known completely. It turns out that one of things that one can know precisely is the energy of the electron in the atom.
If you want to learn more about this at a fairly simple level I would recommend reading Isaac Asimov's "Understanding Physics" or Gerald Feinberg's "What Is The World Made Of?"
Carl Zorn, Detector Scientist (Other answers by Carl Zorn)