A Higgs bundle is a holomorphic vector bundle $E$ together with a 1-form $\Phi$ with values in the endomorphisms of (the fibers of) $E$, such that $\Phi \wedge \Phi = 0$.
The term was introduced by Nigel Hitchin as a reference to roughly analogous structures in the standard model of particle physics related to the Higgs field.
Higgs bundles play a central role in nonabelian Hodge theory.
Let $\mathcal{E}$ be a sheaf of sections of a holomorphic bundle $E$ on complex manifold $M$ with structure sheaf $\mathcal{O}_M$ and module of Kähler differentials $\Omega^1_M$.
A Higgs field on $\mathcal{E}$ is an $\mathcal{O}_M$-linear map
satisfying the integrability condition $\Phi\wedge\Phi = 0$. The pair of data $(E,\Phi)$ is then called a Higgs bundle.
(Notice that this is similar to but crucially different the definition of a flat connection on a vector bundle. For that the map $\Phi$ is just $\mathbb{C}$-linear and the integrability condiiton is $\mathbf{d}\phi + \Phi\wedge\Phi = 0$.)
Higgs bundles can be considered as a limiting case of a flat connection in the limit in which its exterior differential tends to zero, be obtained by rescaling. So the equation $d u/dz = A(z)u$ where $A(z)$ is a matrix of connection can be rescaled by putting a small parameter in front of $d u/dz$.
Analogous to how the de Rham stack $\int_{inf} X = X_{dR}$ of $X$ is the (homotopy) quotient of $X$ by the first order infinitesimal neighbourhood of the diagonal in $X \times X$, so there is a space (stack) $X_{Dol}$ which is the formal competion of the 0-section of the tangent bundle of $X$ (Simpson 96).
Now a flat vector bundle on $X$ is essentially just a vector bundle on the de Rham stack $X_{dR}$, and a Higgs bundle is essentially just a vector bundle on $X_{Dol}$. Therefore in this language the nonabelian Hodge theorem reads (for $G$ a linear algebraic group over $\mathbb{C}$)
where the superscript on the right denotes restriction to semistable Higgs bundles with vanishing first Chern class (see Raboso 14, theorem 4.2).
For a Higgs bundle to admit a harmonic metric (…) it needs to be stable (…).
In nonabelian Hodge theory the moduli space of stable Higgs bundles overa Riemann surface $X$ is identified with that of special linear group $SL(n,\mathbb{C})$ irreducible representations of its fundamental group $\pi_1(X)$.
In the special case that $E$ has rank 1, hence is a line bundle, the form $\Phi$ is simply any holomorphic 1-form. This case is also called that of an abelian Higgs bundle.
The moduli space of Higgs bundles over an algebraic curve is one of the principal topics in works of Nigel Hitchin and Carlos Simpson in late 1980-s and 1990-s (and later Ron Donagi, Tony Pantev…).
N.J. Hitchin, Stable bundles and integrable systems, Duke Math. J. , 54 (1987) pp. 91–114 MR89a:32021 doi euclid; The self-duality equations on a Riemann surface, Proc. London Math. Soc. (3) 55 (1987), no. 1, 59–126, MR887284 doi; Flat connections and geometric quantization, Comm. Math. Phys. 131, n 2 (1990), 347-380, euclid
Carlos Simpson, Higgs bundles and local systems, Publ. Mathématiques de l’IHÉS 75 (1992), p. 5-95, numdam, MR94d:32027
Ludmil Katzarkov, Dmitri Orlov, Tony Pantev, Notes on Higgs bundles and D-branes, (delivered as a lecture by Tony Pantev at winter school at Guanajuato 2013 link) draft pdf
S. B. Bradlow, O. García-Prada, P. B. Gothen, WHAT IS…a Higgs Bundle?, Notices AMS, pdf
Michael Murray, Danny Stevenson, Higgs fields, bundle gerbes and string structures, arxiv/math.DG/0106179
David Baraglia, Cyclic Higgs bundles and the affine Toda equations, arxiv/1011.6421
Around lemma 6.4.1 in
See also
Discussion in terms of $X_{Dol}$ is in
Carlos Simpson, The Hodge filtration on nonabelian cohomology, Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 217{281. MR 1492538 (99g:14028) (arXiv:9604005)
Alberto García Raboso, A twisted nonabelian Hodge correspondence, PhD thesis 2014 (pdf slides)