A scatter plot is a diagram where some entity is measured with two different variables, and the two measurements are plotted on Cartesian coordinates. Scatter plots are widely used in many different fields of study, but they are probably of more use in the social sciences than in the physical sciences.
The purpose of a scatter plot is to be able to see patterns in data that may be more obvious through graphical form than they would be through other means. The purpose of a scatter plot is usually to find (or to not find) correlation. If the question was a simple physical relationship, as to whether lead bricks got heavier as they got larger, there would be no reason to run a scatter plot, except just to find a correlation of 1, which hardly ever exists. The correlation can be done with a mathematical formula, but this often may not capture some of the intricacies that would be readily apparent from a visual examination of a scatter plot.
This is especially important when examining social science data, amongst entities that may be far from equal. For example, if you were examining data comparing alcohol consumption and life expectancy in the United States, a visual examination showing that Utah, Hawaii, and New Hampshire all had high life expectancies while having low, medium and high alcohol consumption rates respectively, would let the viewer know that these states, because of their unique cirumstances, were probably not indicative of the overall trend. And even setting aside these type of considerations, some of the mathematical shapes that scatter plots present are hard to describe simply: for example, in a plot of high school graduation rates and the 2008 election, McCain's strongest support tends to come from states with either very high, or very low, high school graduation rates, giving the graph the shape of a backwards Pac-Man. If there is a technical mathematical term for that, I am not aware of it.
Any two variables that can be operationalized can be plotted, and the results, while sometimes spurious and misleading, can be very thought provoking, and can lead to finding variables that do have a useful correlation between them. It is also very easy to make scatter plots currently, since any office program has a simple functionality for making them.
If reading this makes you as fascinated by scatter plots as I am, you may want to read my blog: