Gauge theories are special and very important classes of relativistic quantum field theories for particles and interactions (e.g. QED, QCD). They are based on the symmetry and invariance properties of the equations during transformations on the variables of these equations.
Just as for the equations of mechanics there are quantities that are conserved such as momentum and energy, and which also derive from symmetries in these equations, these theories automatically produce charges and other numbers that are conserved, associated with elementary particles.
They were introduced by Chen.N Yang and Robert Mills in 1954 to describe the interactions between hadrons.