Graphs are visual representations of data, and can therefore serve a number of purposes. For one, they can summarize data into a form which the human mind can interpret and find patterns in much faster and with greater ease than by looking at the numerical data. Graphs are also used to understand patterns and trends, and verify calculations.
A good statistical graph, therefore, is one that displays statistical data in a way that is useful based on the type of data available. If you are looking for trends over time, for example, then a simple pie chart will not help as much as a time series bar chart or scatterplot, for example. To make sure your graph fits its purpose try to ask yourself what variables are being considered. In the previous example, time is a variable, so time should be along one axis (generally the x-axis along the horizontal). Whatever data you are using to look for trends, those numeric values would be charted against the y-axis along the vertical.
Graphs should always include titles, axis titles and appropriate axis units and increments. When working with multiple sets of data, graphs should also have legends or data labels. Trend lines are also useful in certain types of graphs, such as bar charts and scatterplots. Descriptive statistical data calculations, such as means or correlations, make for nice additions to statistical graphs, to provide more information about the set of data.
Control charts are a very specific type of time series graph, and include not only the data itself, but control limits (and optionally, specification limits) as well. These control limits provide a readily visible indication when something is going wrong in the process you are trying to maintain control of, whether the mean is going too high or too low. While there are many good uses of control charts, keep in mind their ultimate purpose: to see how in control a process is. If over time you expect your data to fluctuate, like in a stock that you hope to go up, or in inventory which you hope to go down, control charts are not appropriate, as being in control of such ideas would be a bad thing.
There are some interesting characteristics of statistical graphs that may help you in your analysis. Most graphs we see in statistics are Normal distributions, that is, the graph is shaped like a bell curve, and a majority of the data falls equally around the mean. The amount the data is cluttered around the mean can be measured with its kurtosis. When the data is not equally distributed around the mean, the distribution may not be Normally distributed, or it may have a skewness to it, making it lean one way or another. Being able to talk to these descriptive characteristics of statistical graphs will help you better compare graphs and better understand differences between such graphs.
In summary, good statistical graphs should display data in a way that is useful and can be readily interpreted and understood. The graphs should include titles and labels or legends so that there is no question what data is being represented, and the axes should have appropriate units and increments to be able to see trends and patterns easily.