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What determines whether a transition-metal complex is high spin or low spin?

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A color-coded periodic table of the elements displays each element's symbol and atomic number, with categories such as alkali metals, noble gases, halogens, transition metals, and actinoid elements distinguished by different background colors. — transition metal Modern version of the periodic table of the elements (printable).

Whether a transition-metal complex is high spin or low spin is determined by the competition between two primary factors: the crystal field splitting energy ( $Δ$ ) and the pairing energy.1

In an octahedral complex, the d orbitals split into two energy levels: a lower-energy set (t_2g) and a higher-energy set (e_g). The number of unpaired electrons depends on how these orbitals are populated:21

High-spin complexes: Occur when the splitting energy ( $Δ$ ) is relatively small compared to the pairing energy. Electrons prioritize occupying separate orbitals to minimize repulsions, resulting in a maximum number of unpaired electrons. This is typically caused by "weak-field" ligands.213
Low-spin complexes: Occur when the splitting energy ( $Δ$ ) is large compared to the pairing energy. Electrons prioritize occupying the lower-energy t_2g orbitals, even if it requires pairing up, resulting in fewer unpaired electrons. This is typically caused by "strong-field" ligands.213

This behavior is most relevant for metal ions with $d^{4}$ through $d^{7}$ configurations, where both spin states are possible. Ligands are ranked by their ability to cause this splitting in the spectrochemical series.23