The dots represent a quantum mechanics calculation of where the electron is likely to be for the lowest energy state. Each dot is a separate calculation for some instant in time. The dots appear randomly, but after many calculations a pattern emerges. In this case the electron is most probably found within a sphere.

The dots represent a quantum mechanics calculation of where the electron is likely to be for one of the orbitals of the second lowest energy state. Each dot is a separate calculation for some instant in time. The dots appear randomly, but after many calculations a pattern emerges. In this case the electron is most probably found within a "dumbbell" shape.