In this post I will explain why the use of area in visualisations should usually be avoided. Circles do have some uses though, so I also offer advice on how to use them.
Visual elements such as area, length and position on a scale can communicate numbers but they do so with different accuracy. A good visualisation should allow the viewer to understand the number presented. The problem is that people do not, in general, judge the area of circles accurately. It is therefore better to use a more accurate visual element, such as a line against a scale.
Circles do have some advantages though which might mean you still want to use them. Their position can communicate other variables. See the github punchcard for an example.
How should circles be sized? Making the diameter proportional to the value tends to overemphasise differences. This is also true of scaling both sides of a rectangle. Using circle area instead, the increase in size of a larger circle is not usually perceived to indicate as large an increase in the value as it should. “people’s judgements of area typically are not proportional to area, but rather to area raised to a power less than one”. 
One idea is to scale the circle to match typical human perception. A possible scaling law is given in Tufte: reported perceived area = (actual area)^x, where x = 0.8 ± 0.3
The problem with this idea is that perception varies greatly with many variables: among people and with the context of the graphic, for example. Tufte recommends that the dimensions of visual elements should be directly proportional to the value represented. The justification is that there is so much variability in perception that it will never be correct, so at least make something right.
The circles should preferably by labelled so that the actual values can be read. A note saying how the size of the circle relates to the value could be useful. It is probably best to acknowledge that area variations will only give a rough qualitative impression of differences in values.
It is even more complicated when you start putting the circles in a bigger graphic, for example neighbouring circles and other features affect how the size of a circle is perceived on a map. 
References and Bibliography
1. Graphical Perception: Theory, Experimentation, and Application to the Development of Graphical Methods
William S. Cleveland and Robert McGill
Journal of the American Statistical Association, Vol. 79, No. 387 (Sep., 1984), pp. 531-554
2. Tufte, 2001: “The Visual Display of Quantitative Information”, p55
3. Judgments of Circle Sizes on Statistical Maps
William S. Cleveland, Charles S. Harris and Robert McGill
Journal of the American Statistical Association, Vol. 77, No. 379 (Sep., 1982), pp. 541-547
4. Influences of Map Context on Circle Perception
Patricia P. Gilmartin
Annals of the Association of American Geographers, Vol. 71, No. 2 (Jun., 1981), pp. 253-258
5. How Numbers Are Shown: A Review of Research on the Presentation of Quantitative Data in Texts
AV Communication Review, Vol. 25, No. 4 (Winter, 1977), pp. 359-409
Circles on maps