What is a 'potential' in the context of physics and gauge symmetry?

I heard someone say that the definition of a gauge transformation is any formal, systematic transformation of the potentials that leaves the fields invariant. What is the definition of a potential in the context of fields (such as QFT)? If possible, in layman's terms.

Note that the notion of gauge theories is much more general than, say, Yang-Mills theory. There is of course a long list of gauge theories that have gauge potentials -- e.g. in the case of Yang-Mills theory, the $$A^a_{\mu}(x)$$ field is the gauge potential -- but it is not a general requirement for a gauge theory.
The non-derivative terms of a field $$\phi$$ in the Lagrangian are often (not always) collectively called its "potential" $$V(\phi)$$. This is because in field theory, the field $$\phi$$ is the analog of coordinates $$q$$ particle mechanics, and there potential functions usually contain powers of $$q$$.In the context of electromagnetism, the field $$A^\mu$$ itself is a potential, called electromagnetic four-potential $$A^\mu=(\phi/c,\textbf{A})$$.