What determines the energies of the electrons surrounding the nucleus of atoms?

That's a tricky question to answer precisely.  The energies are a combination of electrostatic potential (which depends on the charge of the electron and the nucleus and the position of the electron) and kinetic energy (which depends on the mass and speed of the electron) as the electron buzzes about.

By solving the schrodinger equation for the atom, you can determine what the allowed energies are.  Electrons can hop from one energy level to another if the spot is open and it gets (or emits) energy from a photon.

You can describe an electron in a bound system in terms of its quantum numbers.  The energy is mostly a function of its principle quantum number--the solution of the radial part of the schrodinger equation.  But in a multi-electron atom, the energy will also depend to a lesser extent on the electron's angular momentum.  And in the presence of a magnetic field, the energy levels are further split by the direction of the angular momentum (and the spin).  And there are tiny splittings in the energy levels do to relativistic effects like coupling between the electrons' and the nucleus' magnetic moments.

So there's really a LOT to this question, and I can barely scratch the surface in a short answer.