That type of map, where varying sized circles are placed on top of the map features of a given map layer, is generated by using AWhere's 'Map Charts' tool. The Map Charts tool allows you to place either Pie charts or Bar charts on the map. In this case, you would use the Pie chart option, but you would use the values of only one attribute from the map layer...that is what will make the 'Pie' chart just a solid color. (If you were to use more than one attribute to generate the pie charts..the pie charts would have wedges, as you normally recognize as being a pie chart. But in this case, you would just use the one attribute, thus your pie charts only have one 'wedge'...i.e they are solid in color). The varying sizes of the pie charts are determined by the numeric value of that attribute that you are using to generate the pie chart...the larger the number, the larger the circle/pie-chart (relative to all of the other circles on the map).
To learn how to plot Map Charts in AWhere's map window, please access AWhere's help documentation by selecting 'Help' from AWhere's Help menu, and use the 'Search' tab to look up 'Map Charts'. There will be several help-topic pages describing Map Charts, and providing 'How to' instructions for how to make them.