Different arrangements of unobservable fields, can result in congruent observable quantities, and it is the transmutation of one field, to another, that is often referred to as a gauge transformation. Furthermore, the lack of notable change in the observable quantities after a transformation of this type, is referred to as gauge invariance, as the observable physical quantities are symmetrical in both fields.
The power of gauge theories lies partially in their insistence upon rigor, as any change induced at the beginning of a transformation, or in the process, must be canceled out in the end observable quantities.
History
The earliest field theory, is gifted to us via James Clerk Maxwell, who developed the formula forElectrodynamics, but while this contained a gauge symmetry, it went unnoticed.
Following this, Weyl introduced the term gauge, in attempting to explain a potential local symmetry of Electromagnetism, but did so incorrectly, however he and other mathematicians/physicists corrected this later on.