Since you are  asking about Higgs bundles, I can say a few words here. 
These were introduced were introduced by Hitchin in the mid 1980's, although I'm not sure he used this term.  One can look at the introduction to his paper "The self-duality equations on Riemann surfaces" for some of the motivation and background. In brief outline, he starts from a gauge theoretic perspective: the initial data is a pair consisting of a vector bundle with connection on a compact Riemann surface $X$ and additional vector valued $1$-form (the Higgs field) subject to the equations referred to above. One of his main results is that such a pair is equivalent to a pair $(E,\theta)$ consisting of a holomorphic vector bundle and holomorphic section $\theta\in \Gamma(\Omega_X^1\otimes End(E))$
satisfying a suitable stability condition.  One consequence is: 

> Theorem. There is a correspondence between irreducible representations of  $\pi_1(X)$ and stable Higgs bundles of degree $0$. 

Simpson rightly christened (a generalization of) this the *nonabelian
Hodge theorem*. It is indeed a  beautiful and fundamental theorem. It  generalizes an earlier theorem of Narasimhan and Seshadri that corresponds to the case where the Higgs field is zero. 
These objects also make their appearance in Geometric Langlands -- but I'm hardly the right person to discuss these aspects.
I could go on, but I think I'll stop here.