When reading anything about the Max Planck's work we often come across the "oscillation of an electron".

I've just managed to get around to understanding electron orbitals... The dumbbell shape for P orbitals, spherical shapes for S and so on.. (at a very basic level). From what I understand we can't predict with certainty where the electron will be.. This sounds to me like a random movement within the specified orbital. In that case, what's this oscillation I keep reading about?

Even though you can't predict a location, since they just don't travel in classical trajectories, there is an oscillation frequency associated with the electrons. They are confined in a potential well with a certain energy. When they make transitions, they can be seen as acting like electric dipoles (or, more rarely, magnetic or electric quadrupoles)

you mean when they jump between energy levels?

Yes. Many transitions emit radiation just like a dipole.