Our most popular applet, by far, is The ε - δ Applet, showcasing the mathematical definition of a limit. To emphasize to your students how important the definition is, we recommend the surprising Limits via Estimating Slope applet. Although the three functions look identical at first glance, they look quite different as you zoom in.

For a possibly open problem that doesn't require a doctorate to begin, see Henri's φ-Curve.

Finally, we recommend Geometry Playground, a straightedge and compass application in your choice of Euclidean or five non-Euclidean (spherical, projective, toroidal, hyperbolic, and Manhattan) geometries. This represents a year and a half of our work, and we hope that math teachers and students find it useful.