For collinear spin, where the spin quantization axis is along $z$, one has
\begin{align}
S_z = \frac{{N_\alpha}-{N_\beta}}{2}
\end{align}
Multiplicity $(2S+1)$ can then be specified given a particular number of up $(\alpha)$ and down $(\beta)$ electrons.
In unrestricted Hartree-Fock (UHF), the $\alpha$ and $\beta$ molecular orbitals $(C)$ aren't restricted to be the same spatially. This allows for greater variational flexibility in the self-consistent-field (SCF) procedure for finding open-shell solutions, since spin symmetry constraints are lifted.
In this case, we solve the Pople-Nesbet equations — again using different orbitals for different spins
\begin{align}
F^{\alpha} C^{\alpha} = SC^{\alpha}{\epsilon}^{\alpha}
\end{align}
\begin{align}
F^{\beta} C^{\beta} = SC^{\beta}{\epsilon}^{\beta}
\end{align}
The UHF energy can be expressed as
\begin{align}
\begin{split}
E_{\mathrm{UHF}} = \sum\limits_{i}^{N_{occ}^{\alpha}}h_{ii} +
\sum\limits_{\overline{i}}^{N_{occ}^{\beta}}h_{\overline{ii}} +
\frac{1}{2}\sum\limits_{ij}^{N_{occ}^{\alpha}}\mathcal{J}_{ij}^{\alpha\alpha}+
\frac{1}{2}\sum\limits_{\overline{ij}}^{N_{occ}^{\beta}}\mathcal{J}^{\beta\beta}_{\overline{ij}}\\+
\frac{1}{2}\sum\limits_{i\overline{j}}^{N_{occ}^{\alpha\beta}}\mathcal{J}^{\alpha\beta}_{i\overline{j}}+
\frac{1}{2}\sum\limits_{\overline{i}j}^{N_{occ}^{\beta\alpha}}\mathcal{J}^{\beta\alpha}_{\overline{i}{j}}-
\frac{1}{2}\sum\limits_{ij}^{N_{occ}^{\alpha}}\mathcal{K}_{ij}^{\alpha\alpha}-
\frac{1}{2}\sum\limits_{\overline{ij}}^{N_{occ}^{\beta}}\mathcal{K}_{\overline{ij}}^{\beta\beta}
\end{split}
\end{align}
For a given geometry, the benefit of performing separate SCF calculations for each spin state is orbital optimization. However, significant spin polarization can lead to spin contamination, where your single determinant solution is no longer an eigenstate of the total spin-squared operator $\mathcal S^2$.
Deviations from the ideal spin state is observed in the following expectation value
\begin{align}
\langle{\mathcal{S}^2}\rangle_{\mathrm{UHF}} = S_z(S_z + 1) + N_{\beta} - \sum\limits_{i\overline{j}}^{N_{occ}} {\left| \mathbb{S}_{i\overline{j}} \right|}^2
\end{align}
Note that when the $\alpha\beta$ molecular orbital overlap term ${\left| \mathbb{S}_{i\overline{j}} \right|}^2$ does not cancel with $N_{\beta}$, one can encounter a situation where $\langle{\mathcal S^2}\rangle_{\mathrm{UHF}} > S_z(S_z + 1)$. The amount $\langle{\mathcal{S}^2}\rangle_{\mathrm{UHF}}$ can deviate from the exact value while still retaining a viable wavefunction is variable and depends on the how the reference determinant will be used. In general, one would like to have an $\langle{\mathcal{S}^2}\rangle_{\mathrm{UHF}}$ no greater than $10-15\%$ of the target spin.
Spin contamination can of course be ameliorated or avoided via spin recoupling by expanding the wavefunction in a basis of determinants as done in configuration interaction and related methods. Restricted open-shell SCF methods that conserve spin exist, however, specifics regarding their construction and use can be found elsewhere in the literature.
In standard implementations, the principles of the UHF method can be applied to unrestricted Kohn-Sham (KS-)DFT as well. Though, the concept of spin in KS-DFT is a bit ambiguous and there are plenty of discussions on its validity/practicality. Here are just a few papers on the topic:
https://doi.org/10.1080/00268976.2017.1393573
https://doi.org/10.1002/qua.21423
https://doi.org/10.1021/jp803064k
https://doi.org/10.1002/qua.560560414
For routine energy calculations or structure optimizations, you're most often fine with using your preferred density functional approximations as a starting point to model the chemical problem of interest.
