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The Doppler Effect is an easily observable phenomenon where the perceived frequency (or wavelength) of a wavetrain is altered by the observer's motion relative to the source of the signal.
When moving towards the source, the frequency will appear to be higher and the wavelength shorter; when moving away from it, the frequency appears lower and the wavelength longer. Red shift or redshift is the observed reduction in frequency of electromagnetic radiation perceived when the source of the radiation is moving away from the observer. A corresponding increase in frequency observed from approaching sources is known as blue shift or blueshift.
We notice this in ordinary life when listening to a rapidly moving vehicle approach and then depart from our listening position (especially if it has a siren!). The pitch is higher as it approaches and lower as it leaves. In the case of vehicles with sirens, this change to a decreasing pitch is often accompanied by a sigh of relief — the "Doppler Sigh". Radar and laser speed guns generally use shift measurement of reflected radiation to ascertain speeds of moving objects.
In astronomy, this effect is combined with theories about the structure of the Universe to determine the distance to sources of light and other electromagnetic radiation. Since we know the characteristic frequencies of various radiation-producing phenomena, we can compare the spectrum we observe and see how far it is "red shifted" (in the case of objects moving away from us) or "blue shifted" (for those approaching). The red shift is basically proportional to the distance an object is away from us, due to the continuous expansion of the universe.
Astronomy uses the frequency shift of various known marker lines in stellar emissions to determine the relative rate of approach (rare but existent) or departure of objects at any distance from the observer. It was the conclusion that distance was proportional to red shift (the further away an object, the greater its speed of departure) that caused Edwin Hubble to propose that the Universe is expanding.
 "Quantized redshift"
Quantized redshift refers to the clustering of galaxies in concentric shells away from Earth at periodic distances. Evidence for it is flimsy at best, and consequently it is largely ignored by most experts. However, it is used by creationists and geocentrists to argue that Earth is at the center of the Universe. Creationists have taken the data to indicate that our galaxy is at the "centre" of the universe.
Starting in 1973 and up to the late 1990s, some studies indicated the possibility for a periodicity in galaxy clusters depending on the statistical methods used for data analysis. Starting in the late 1990s, and continuing to the present day, the redshifts of a much larger numbers of galaxies were measured, providing significantly more data. Recent studies indicate no evidence or weak evidence. Another survey provided weak evidence, but a subsequent re-analysis of the same data brought that conclusion into question, explaining the observations as an effect of the statistical techniques used to derive it.
It can safely be said that evidence for the existence of a quantized redshift is hardly convincing. It appears that the only people who take it seriously are those who reject conventional cosmological models.
 Alternate explanations
 Interstellar reddening
If a gas or dust cloud lies between an astronomical object and the observer, the light is "reddened" somewhat. This effect is markedly different from a red shift, however. In a red shift, the spectral lines appear at lower frequencies. Interstellar reddening doesn't effect the frequency of the spectral lines, it merely reduces the intensity of higher-frequency light, leaving more lower-frequency ("redder") light to arrive at the detector.
In other words, reddening due to material between the source and the observer does not explain the red shift seen for very distant objects; only motion of the source away from the observer can explain the red shift.
 Tired light
To explain the universal greater-redshift-with-distance Hubble was observing, he came up with an explanation called tired light. It was, quite simply, the notion that light loses energy (and therefore gets longer in wavelength) when travelling over great distances, just because.
Even leaving general relativity aside, there are problems with just about every model of tired light that has ever been proposed, as summarized in this article on the Other Wiki.
The observed frequency can be calculated as follows:
where V rel is the relative closure speed of the source and observer.