Hitchin equations were the first solution to the hyperkahler moduli space of the Higgs bundle (see here). Since then, it has appeared many times in the physics of gauge theory. In fact, the reduction of $\mathcal{N} =4$ SYM to a two-dimensional gauge theory has target space (at low energies) which is just he Hitchin's moduli space of a Higgs bundle $\Phi$.
Let's see what are the Hitchin equations. Take a Higgs bundle on a smooth Reimann surface $C$, then we have
- A holomorphic vector Bundle $E$
- Holomorphic Higgs field $\Phi$. This is but the holomorphic section of the endomorphism bundle $End(E) \otimes K_X$ where $K_X$ is the canonical bundle of $X$.
Higgs bundle $\Phi$ is one-form $C$ which takes value in the adjoint representation (which is the adjoint bundle of G-bundle $E$).
Now, we take the bundle and define the complex connection $\mathcal{A}$ over the $G_\mathbb{C}$ bundle
$$\mathcal{A} = A+i\phi$$
$G_{\mathbb{C}}$ is given by the complexification of $E \to \mathbb{C}$ and the structure group $G \mapsto G_\mathbb{C}$. The complex curvature $\mathcal{F}$ is given by
$$\mathcal{F} = d\mathcal{A} + \mathcal{A} \wedge \mathcal{A}.$$
We can get the real and imaginary parts of this curvature as
$$\text{Re}\mathcal{F} = F-\Phi \wedge \Phi$$
$$\text{Im}\mathcal{F} = D\Phi$$
The Hitchin equations are now
$$\mathcal{F} =0 \\ D \star \Phi=0$$
where $D \star \Phi = D^-_Z\Phi_z+D_z\Phi^-_z$, for $z \in \mathbb{C}$, so
$$D^-_z = \partial^-_z+ [A^-_z, \cdot]\\ D_z=\partial_z + [A_z,\cdot]$$
For a $\mathcal{N} =4$ SYM on a four manifold $\Sigma \times \mathbb{C}$ where $\Sigma$ is a 2-manifold and is very large than $\mathbb{C}$. $\mathbb{C}$ is just the Riemann surface. Now, we reduce the four dimensional supersymmetric gauge theory on $\Sigma \times \mathbb{C}$ to $\Sigma$. That should make clear why $\Sigma$ is larger than $\mathbb{C}$ and we get the effective field theory on $\Sigma$. This two dimensional theory will be the SUSY $\sigma$-model. The presence of such sigma model is a fortunate situation. The target space of this sigma model is the space of classical supersymmetric vacua that we get while the compactification (this is pretty known in physics). Evidently, what we get is that the target space is space of Hitchin hyperkahler moduli space of the Higgs bundle $\Phi$ that we just discussed. This moduli space is widely studied in the physics literature and usually represented by $\mathcal{M}_H(G,\mathbb{C})$ where $G$ is the structure group of $E$-bundle. So basically the Higgs bundle is given by the pair of a complex connection and Higgs bundle $(\mathcal{A},\Phi)$. And we are interested in solutions in the map $\Sigma \to \mathcal{M}_H$.
