Before Niels Bohr:
in the classic physics, the world is described by a few equations that govern a set of continuous quantities or fields. It assumed that there is a smaller portion of anything, but maybe out of our measurement capability to see it. This continuum picture of world, which roots in the primary works of Newton, works very well in the daily life. But, there are some exceptions, and one of them is incapability of classical physics in explaining the very stable state of atoms.
The well-accepted and simple picture of classical physics of an atom is a set of electrons who rotate around a positive nucleus which consists of protons. The problem of this picture is that this model cannot be stable, as a rotating charge radiates its energy and will collapse in the nucleus in a very short period of time. Despite this prediction, atoms in the real world are very stable, and are the basic elements of almost all materials.
After Niels Bohr:
Bohr tried to explain this contradiction by introducing a new concept in the physics: quantum. He claimed that electrons could only rotate on certain discrete orbits around the nucleus. The special thing about these orbits is that electrons on them save energy, and don't radiate energy. He was able to determine these special orbits using the Max Planck's radiation equation. As long as an electron stays on one of these orbits, Bohr model insures its stability.
The question was how electrons set on these orbits. According to Bohr, electrons are allowed to move from one orbit to another one by giving or taking the associated energy difference between two orbits. These movements result in release or absorption of discrete energy units of radiation, i.e., photons.The probability that an electron moves downward is high when the electron goes to higher orbits. In other words, an excited atom tends to release its energy in the form of photons and come back to its lower energy.
Based on this interpretation, Bohr was able to explain the lines in the spectra of gases. These lines represent transitions of electrons to and from various orbits. As the orbits are finite, the possible lines are also finite, and usually they are unique for each atom. The simplest atom, Hydrogen atom, follows accurately Bohr's predictions.
What Niels Bohr missed:
There are a few semi-classic assumptions in the Niels Bohr model. First, the orbits in this model are classical circles and ellipses. Second, he used the correspondence principle in the calculations of the transition probabilities.