What does it mean for an atomic spectrum to be discrete, and how does it relate to energy level spacings?

All atomic spectra are discontinuous/discrete line spectra.

We have consistently defined orbitals as #1s#, #2s#, #2p#, etc., where the number is the quantum number #n#, but #n# is never a decimal. #n# is #1,2,3,...# only.

Since an energy level is quantized, meaning it is discrete in integer steps, the electronic transitions that emit light in the form of these line spectra also give discrete lines.

Note that the energy levels get closer and closer together the higher the energy level is, but they are still discrete nevertheless.

To emit light, electrons drop from higher-energy orbitals to lower-energy orbitals to relax their energy state and release energy. The electron never stops halfway in between energy levels, and the energy jump it makes must span the full energy gap. This energy is observed as light energy.

The higher the energy gap #DeltaE# of the electronic relaxation between #n = A# and #n = B#, the higher the frequency #nu# and the lower the wavelength #lambda# of the light that you see.

Red is closer to #"700 nm"# and violet is closer to #"200 nm"#. The UV-Vis spectrum is #"200~400 nm"# (UV) and #"400~700 nm"# (Vis).