A Conformal Map is a function from the complex plane to itself that preserves angles. That is, if two curves in the complex plane meet in a certain angle, then the images of those curves under the map meet in the same angle. Any complex analytic function is conformal. The purpose of the Conformal Maps Gallery is to help visualize the mapping properties of complex analytic functions and of other functions from the complex plane to itself. The difficulty, of course, is that the graph of such a function exists in four dimensions. So, instead of trying to look at the graph, we look at a grid in the plane and then at the image of that grid under the map. The Action menu gives you a choice of three types of grid: Cartesian, Polar, and Polar Conformal. With a Polar or Polar Conformal grid, the input grid is a "wedge" in the plane, rather than a rectangle. (A Polar Conformal grid is the image of a cartesian grid under the conformal map exp(z).) The range of values displayed in the grid are given by parameters umin, umax, vmin, and vmax; these parameters can be set using the "Set Parameters" command in the "Settings" menu.
When you select an Exhibit from the Conformal Map menu, the default view of the exhibit shows the input grid on the left-hand side of the window; on the right is the image of that grid under the map. Grid lines are color-coded to help you see which parts of the input grid map to which parts of the image. Furthermore, a Create animation runs in which a line sweeps across the input grid first in one direction and then the other, and at the same time the image of this line sweeps across the image on the right-hand side of the window. (To see this animation again, select the "Create" command from the "Actions" menu.)
You can add additional lines and circles to the domain. Choose one of the commands "Choose Interval by Mouse," "Choose Line by Mouse," or "Choose Circle by Mouse" from the "Actions" menu. Then click-and-drag the mouse on the left side of the window. As you drag the mouse, you draw a figure on the domain. When you release the mouse, the image of the figure is added to the right side of the window.
The complex plane (plus a "point at infinity") can be identified with the Riemann Sphere, which exists as the surface of a sphere in three dimensions. The "Actions" menu gives you the option "View Image on Riemann Sphere." With this option, the view of the image is three-dimensional, and the image of the map lies on the surface of the unit sphere.
If you want to see only the image and not the input grid, the "Show Both Domain and Image" option in the "Actions" menu can be turned off. In that mode, you can use the mouse to temporarily switch between the view of the input grid and the view of the image: Shift-right-click on the window. As long as you hold down the mouse button, the input grid will be shown; when you release the mouse button, the window will go back to showing the image. (On Mac OS, you can left-click while holding down the shift and command keys for the same effect.)