Bohr's model of atom
Since this is pretty well covered in countless number of books and other sources I will try to just emphasise the key points.


The emission spectrum of atomic hydrogen was well known at that time, the only problem was that no one was able to explain it. :D Where does all this lines come from, i.e. what physical process gives rise to such discrete spectrum?
The wavelengths of spectral lines were known to be very well predicted by the Rydberg formula, but again no one was able to explain why this formula works, i.e. what are its physical justifications.
Bohr model explained discrete spectrum as follows: spectral lines are due to electron transitions between stationary orbits. Bohr was also able to determine the actual energy spacing between levels in hydrogen atom and explain the reason for the structure of the Rydberg formula.
However, Bohr failed to predict the lines in the emission spectrum even for helium atom. It happened simply because Bohr model is physically  wrong. Electrons do not orbit around the nucleus.