The experiment by Rutherford showed that an atom has a small, heavy nucleus at its center, with electrons going around it.
In electromagnetism, however, a charge in circular motion emits electromagnetic waves. Going around while giving out energy, the electron traces a spiral into the nucleus. The calculation has it falling in within about 10^{-11} seconds.
Real atoms do not collapse. What is more, the light an atom emits comes at particular frequencies only. If it were emitted while falling in, the frequency ought to change continuously. Classical electromagnetism does not hold inside an atom.
Bohr began by placing two things. One is that the electron goes around only on particular, discrete orbits. The other is that while it is on such an orbit it emits no electromagnetic waves.
These orbits are called stationary states. Why nothing is emitted is not explained. They were placed to fit the facts that the atom does not collapse and that the light coming out is discrete.
Whether an assumption that was merely placed is right is settled by what comes out of it. From here Bohr calculated the wavelengths of the light hydrogen emits, and matched them against values that had already been measured.
What decides which orbits are allowed is the quantum condition. Bohr himself wrote it with angular momentum, but using the de Broglie wave makes the meaning visible.
Take the electron as a wave. If crest and trough do not line up when it comes back around the orbit, it cancels itself out. The only case that remains is the one where a whole number of waves fits around the circumference.
With the radius r and the wavelength \lambda this is 2\pi r = n\lambda. Putting in \lambda = \dfrac{h}{mv} gives mvr = \dfrac{nh}{2\pi}, which is the quantum condition of Bohr itself.
The Coulomb force on the electron is the centripetal force of the circular motion. Solving \dfrac{ke^2}{r^2} = \dfrac{mv^2}{r} together with the quantum condition fixes the radius. r_n is proportional to n^2, and the innermost r_1 is 0.53 \times 10^{-10} m.
The energy is the sum of the kinetic and the potential energy, and takes the form E_n = -\dfrac{E_1}{n^2}, where E_1 is 13.6 eV. It is negative because the infinitely separated state is taken as 0, which means that the energy of a bound electron is lower than that.
The larger n is, the further out the orbit spreads and the more the levels crowd together. Take n without limit and the energy approaches 0, which is the boundary where the electron comes away.
An electron can move from an outer orbit to an inner one. When it does, the energy left over goes out as one photon.
The energy of the photon that comes out is the difference between the two levels itself, h\nu = E_n - E_{n'}. This is called the frequency condition. Since the levels are discrete the differences are discrete too, so the frequency of the light coming out is discrete as well.
The measured result, that particular lines come out rather than continuous light, is explained by this. In the next article the actual wavelengths are obtained from these differences and matched against the spectrum of hydrogen.