As mentioned in the last post, the Bohr atom is not correct, quantum mechanically speaking. It does, however, do an excellent job in modeling the simplest atom, that of hydrogen. Hydrogen is the lightest element, consisting of one proton with one electron in orbit. With the Bohr atom model, we know that the orbiting electron can exist in various discrete orbits, corresponding to different energies. In addition, we know that when the electron jumps between these levels, it emits or absorbs a photon. Finally, we know that the energy of a photon is proportional to its frequency, which is related to its wavelength or color. Putting all this together, we can predict that a hydrogen atom will emit or absorb very specific colors of light.
We are now talking about "spectral analysis." The OED (do you like the use of the OED, mom?) defines spectrum in a couple of ways, but I'll print two of them here. First, a general definition for physicists: "An actual or notional arrangement of the component parts of any phenomenon according to frequency, energy, mass, or the like." Physicists often talk about an energy spectrum, a frequency spectrum, etc., and what we mean is exactly the definition given by the OED - breaking up some group or data set into its components.
A second definition is this: "The coloured band into which a beam of light is decomposed by means of a prism or diffraction grating. Also, a dark band containing bright lines produced similarly; such a (coloured or dark) band, or the pattern of lines in it, as characteristic of the light source; hence, the pattern of absorption or emission of light or other electromagnetic radiation over any range of wavelengths exhibited by a body or substance." Now this is exactly what I'm talking about. The general idea is familiar - anyone who has seen a rainbow has seen light broken up into its various colors. When applied to the hydrogen atom, the spectrum is the characteristic colors of light that can be emitted or absorbed. Using the Bohr model, we can predict which wavelengths can interact with hydrogen (this may be the subject of another quantitative post).
At right is an image of a tube filled with hydrogen that is being excited by high voltage (taken from here). The light passes through grating to separate the spectral lines, with the result being the smaller lines shown to the right. All hydrogen, anywhere in the universe, will emit these colors when excited.
And now we finally get back to dark matter. In the first post, I talked about how we can tell how fast the galaxies are spinning using redshift or the Doppler effect. That's because any hydrogen in those distant galaxies will have the same spectrum as hydrogen on Earth, which means the hydrogen in the distant galaxy is emitting the same wavelengths of light that we can measure here on Earth. As discussed in the Doppler effect posts, since light is a wave, its wavelength will be shifted via the Doppler effect depending on the speed of the source. Because we can measure the relationship between the different lines, we can use the observed shifts to deduce the rotational speed of the source galaxy (one side spins away from us, one side spins towards us). And voila, now we know that the galaxies are spinning too fast and that there must be matter we aren't seeing (well, we would see that if we understood Newtonian gravity, which will be the topic a future post, I'm sure).
The above plot taken from the website of an MIT experiment, FIRE, shows the redshift for three real objects. Why is it called "redshift?" Most objects in the universe are moving away from us, and when the source of light is moving away, the light shifts towards the red end of the spectrum (or towards lower energy).