This software was designed to compute corresponding circle packings in a chosen closed curve and the unit circle. The purpose of this algorithm is, surprisingly, to approximate conformal mappings, whose existences are guaranteed by the Riemann Mapping Theorem. The best introduction to these concepts can be found on the Wikipedia page, which even includes a brief description of the algorithm in this program.
The usage of this program is very simple. With 'Place Circs' selected, you can place a collection of congruent circles on the grid in a hexagonal configuration, and if it is simply connected this program will compute and display the corresponding packing inscribed in the unit circle.
More fun is that you can select Draw Curve, and draw a closed curve in the grid. The program will automatically place each circle whose center lies inside that curve. You can then use the slider to give you finer and finer grids, and thus, display better approximations of the conformal mapping from the curve you drew to the unit circle!
The algorithm is iterative, and generally for more circles you will need more iterations to get an accurate looking result. The preset of 120 iterations is good for smaller pictures, but you may need to go up to a few thousand for large ones.