How do interactions between nuclear spins split NMR spectral lines?

In nuclear magnetic resonance (NMR) spectroscopy, the splitting of spectral lines is primarily caused by an indirect interaction between nuclear spins mediated by electrons, a phenomenon known as spin-spin splitting.12

While direct magnetic dipolar interactions between nuclei are typically averaged out by molecular motion in liquids, this indirect interaction persists and splits the resonance line of a specific nuclear spin into multiple components.1

The pattern of this splitting follows the $n+1$ rule: a set of equivalent nuclei is split into a multiplet of $n+1$ peaks by $n$ adjacent, nonequivalent nuclei. For example, in the molecule bromoethane, the three equivalent hydrogen atoms of the ${\text{CH}}_3$ group are split into a triplet by the two nonequivalent hydrogens on the adjacent ${\text{CH}}_2$ group, while the ${\text{CH}}_2$ hydrogens are split into a quartet by the three hydrogens on the neighboring ${\text{CH}}_3$ group.2

These splitting patterns, along with chemical shifts and peak integration, provide essential information for determining the molecular structure of organic compounds.2