It's a reference to a particular kind of splitting of the energy levels of electrons in atoms, due to interactions between the electrons and the nucleus.

Basically, as more and more precise measurements of the energy levels of electrons in atoms were made in the 1920s, 30s, and 40s, physicists kept finding that energy levels that were thought to be degenerate (i.e., multiple states with the same energy) were actually split into multiple, closely spaced levels. The original quantum model was the non-relativistic Schrodinger equation as applied to the atom. Then it was found that electron energy levels that were degenerate in that model were actually split into multiple levels because of the effects of electron spin and certain relativistic corrections; this splitting was called "fine structure". Then it was found that there was even further splitting, of energy levels that were degenerate in the fine structure model, due to interactions between the electron and the nucleus; this further splitting was called "hyperfine structure".

More here:

https://en.wikipedia.org/wiki/Hyperfine_structure