Most people living in California have heard about the ``Richter Scale'' and have at least a vague idea that it is used to measure the sizes of earthquakes. Most people that I know, though, have some misconceptions about the Richter Scale -- for example, someone once asked me if they could see where I kept my Richter Scale! Also, there are some important differences between magnitude, energy, and intensity that need to be discussed.
C.F. Richter at Caltech invented the idea of earthquake magnitudes in 1935 as a way to compare earthquakes. He was into astronomy and knew that astronomers used magnitude scales to compare the brightnesses of stars, so he adapted the idea for seismology. Richter based his scale on the way ground motion was recorded by a specific type of seismometer that was very common back in the 1930's, called a Wood-Anderson seismometer. It is very important for you to realize here that the Richter Scale is completely arbitrary; it was made up by Richter.
Basically, if you know how far away an earthquake is from your station, and you have a record from the earthquake, you can calculate its Richter magnitude. You do that by measuring the maximum amplitude of the shaking recorded by the W-A instrument, taking the logarithm of that amplitude, and adding in a number to take into account distance from the station. The key thing here is that the magnitude scale is logarithmic; for every one full point change in magnitude, the amount of shaking recorded by a seismometer will go up by a factor of 10.
Richter magnitude is useful, but limited; it is only defined for local earthquakes. Other magnitude scales have been developed to handle earthquakes that are distant from the seismometer making a given magnitude estimate, and these have been made to give magnitudes which are basically similar to Richter magnitudes. However, all of these scales have the same two fatal flaws. First, all of them become inaccurate at large magnitudes, and in fact, above about magnitude 8 or so, the magnitudes just don't get bigger (even though the earthquakes do). Second, these magnitude scales are all empirical; they don't actually tell you anything about the physics of the earthquake itself.
Recently, a new magnitude scale has been developed which has neither of these flaws: the moment magnitude scale. Unlike all the other kinds of magnitude, moment magnitude tells you something about the physical size of the earthquake. Moment magnitude really is physically meaningful. Also, moment magnitude is never overwhelmed by large earthquakes, and so it is possible to get meaningful estimates of magnitude even for humungous quakes. As an example of this, recently people have gone back and recomputed magnitudes for earthquakes such as the 1964 Alaska earthquake. Previously, the magnitude had been given as 8.6 - pretty damn big, but nothing compared to the 9.2 which is now accepted (remember that magnitude scales are logarithmic). Moment magnitudes are now the accepted magnitude among seismologists, and are usually the numbers given to the press.
Another way of looking at the size of earthquakes is to figure out how much energy they release. Some rules of thumb have been found to compare magnitude to energy, and it has been found that for each one point magnitude increase (say from a 4 to a 5), 32 times as much energy is released. If one jumps from a 5 to a 8, the energy goes up by 32 x 32 x 32, which is almost a factor of 33,000 -- but don't worry. While the total energy goes up that much, it does so not because the ground shakes 33,000 times harder, but instead because large earthquakes release energy for much longer and over a much wider area than do smaller earthquakes.
Finally, there is another way of looking at the strength of earthquakes, which depends not on records of earthquakes but on how the earthquake was perceived by people and how much damage is done. This is called intensity, and is described using the Mercalli Scale. While there will be very little variation in magnitude estimates for a given earthquake, intensity measurements can (and do) vary widely. Intensities vary based on the distance from the earthquake, what the person making an intensity report was doing at the time of the earthquake (intensity would be lower from someone who was air-guitaring to Van Halen than from someone sitting quietly playing a viola, for example), by what kind of building they were in, what kind of soil they were on, etc. Intensities are inherently subjective, but can be of use to engineers who try to build earthquake-resistant buildings, for example.