In the early part of the 20th century, experiments by Ernest Rutherford and others had established that atoms consisted of a small dense positively charged nucleus surrounded by orbiting negatively charged electrons. However classical physics at that time was unable to explain why the orbiting electrons did not spiral into the nucleus.
The simplest possible atom is hydrogen, which consists of a nucleus and one orbiting electron. Since the nucleus is positive and the electron are oppositely charged they will attract one another by [coulomb force]?, in much the same way that the sun attracts the earth by [gravitational force]?. However, if the electron orbits the nucleus in a classical orbit, it ought to emit electromagnetic radiation (light) according to well established theories of electromagnetism.
If the orbiting electron emits light, it must lose energy and spiral into the nucleus, so why do atoms even exist? What's more, the spectra? of atoms show that the orbiting electrons can emit light but only at certain frequencies. This made no sense at all to the scientists of the time.
These difficulties were resolved in 1913 by Niels Bohr who proposed that the orbiting electrons:
These assumptions explained many of the observation seen at the time, such as why spectra consist of discrete lines. Assumption 4) states that the lowest value of n is 1. This corresponds to a smallest possible radius (for the mathematics see Ohanian?-principles of physics or any of the large, usually American, college introductory physics textbooks) of 0.0529 nm. This is known as the [Bohr radius]?, and explains why atoms are stable. Once an electron is in the lowest orbit, it can go no further. It cannot emit any more light because it would need to go into a lower orbit, but it can't do that if it is already in the lowest allowed orbit.
The Bohr model is sometimes known as the semiclassical model because although it does include some ideas of quantum mechanics it is not a full quantum mechanical description of the atom. Assumption 2) states that the laws of classical mechanics don't apply during a [quantum jump]? but doesn't state what laws should replace classical mechanics. Assumption 4) states that angular momentum is quantised but does not explain why.
In order to fully describe an atom we need to use the full theory of quantum mechanics, which was worked out by a number of people in the years following the Bohr model. This theory treats the electrons as waves, which create 3D standing wave patterns in the atom. (This is why quantum mechanics is sometimes called wave mechanics.) This theory considers that idea of electrons as being little billiard ball like particles that travel round in orbits as absurdly wrong, instead electrons form probability clouds. You might find the electron here with a certain probability; you might find it over there with a different probability. However it is interesting to note that if you work out the average radius of an electron in the lowest possible energy state it turns out to be exactly equal to the Bohr radius (although it takes many more pages of mathematics to work it out).
The full quantum mechanics theory is a beautiful theory that has been experimentally tested and found to be incredibly accurate, however it is mathematically much more advanced, and often using the much simpler Bohr model will get you the results with much less hassle. The thing to remember is that it is only a model, an aid to understanding. Atoms are not really little solar systems.