How to choose the appropriate graphic representation (plot) method for the results? Do you use a rule of thumb when deciding the type of the chart/plot (box plot vs. scatter plot) you use when you want to represent the results of your research/simulations?

I would say the answer is "choose the one that represents the data best" but still there must be some type of rule where you surely don't use a given plotting method but use another one.

Or is my understanding of the purpose of plots completely wrong. Hope the more experienced scientists here provide some guidelines
Data visualization is an entire scientific field of its own. A good introduction to the basic principles can be found in the works of Edward Tufte, whose book on the visual display of quantitative information is a classic in the area.

The basic principle, however, is that you want the important information of the chart to leap out to the eye, and as such the dimensionality of your presentation and the dimensionality of the presentation method should be matched as closely as possible.  Likewise, the features of different elements of a graph should be adjusted to put the emphasis on the most important ones, while not concealing the others.  

For a simple example, consider a simple one-dimensional data set, where there are a set of conditions, each of which produced some distribution of values.  


If the conditions are qualitatively different or vary along many dimensions, the data should probably be presented as a bar graph
If they are quantitatively arranged along one dimension they should be presented as a line graph.
If they are arranged along two dimensions, the should be presented as multiple lines on a single graph.  


The type of error bars also depends on the nature of the distributions:


If the distributions are small compared to the effect size, such that they do not significantly affects interpretation of the data, then error bars should be made small and subtle, so that the information is present but the eye is drawn to the means.
If the distributions are large but regular, they can be presented via visually prominent error bars.
If the distributions are large and highly irregular, then a more complex representation like a box-and-whisker graph is better.


This is just scratching the surface, of course, but hopefully gives an illustration of the principles of how to think about data visualization...