A short answer of mine.
First whats potential, before whats gauge potential.
A potential is the variable: energy. ( — Energy in its various forms eg per unit charge or unit mass, known as; electric potential vs gravitational potential ).
All energy are integration of vector-field quantity, so Force is the vector field quantity whose integration is energy.
We say; Force is the potential gradient of energy or potential. ( — and this potential gradient which is a derivative of a higher variable, it could be both negative and positive)
Then we say also, in Physics, Force per unit mass or unit charge? Like we did that for energy. So this time you say electric field vector or gravitational field vector. Like Force was positive or negative gradient (of energy), Field (a force or vector per unit charge or mass) is negative or positive potential gradient.
Potential was, as said above, energy per unit mass or charge. So, we see that, in defining the higher quantity energy or potential (higher therefore closer to action, hence more fundamental or unified) we have to INTEGRATE the lower variable, here, Force, Field or (Any ) 3-vector. This entails therefore arbitrariness into the Physical solution when we solve for these quantities. These physical problems, as they involve differentials or integration, leads to a differential equation. Under further suitable physical conditions called eg laws of nature or physics, become whats called a wave-equation or for particles, equation of motion. We can say equation of motion for particles or equation of motion for waves if they are separate.
Now that we understand what are potential, field, vector and gradient and integral in relation to each other, comes requirements called as symmetry or laws of nature or laws of physics or in simple, boundary conditions to these differential equations known as, wave equation or equation of motion of particles OR waves. ( — which are separate so far )
These equations constrained by the conditions or restrictions which are attributes of physical observation, must therefore unite these variables (potential, field) into one entity which would satisfy the wave or particle equations of motion, the differential equations of motion in PARTICULAR ways only, known as Laws of Nature or Physics. So they become, from their 3–vector or scalar attributes, 4–vectors (or still higher, Tensors).
One word of concept, the potential is also a field but a scalar. So instead of (potential, field) we can write (scalar, 3–vector) components of the field, that is, (scalar potential, vector potential) which as you can see in a 4-dimensional world; 1+3 = 4 components.
Then comes the conditions or inter-relations between these vector and scalar and their differential and integrals. ( — again known as a theory or equation of motion)
These conditions have an arbitrariness built into them, puchho kyon? Munna bhai says: because there are integration. How did Munna bhai figure that out?
He was in a bar dancing with a lady Physicist from Munich, who had come to Mumbai for a conference in Particle Physics.
The method-of-integration has a constant of integration, any one?
In going from lower variables such as velocity and momentum we gradually integrate them to find the higher, closer-to-action or fundamental variables, and incurr arbitrariness into our understanding.
But not all of these solutions are physical in nature, they are merely invalid mathematical solutions even if not necessarily trivial solutions. To remove these arbitrary ness is whats called Gauge Condition. That is, to chose the right scalar and vector potential among all solutions, only that, which would be Physically correct.
Therefore a gauge or gauge potential ( — or a gauge field scalar or gauge field vector) are what’s more fundamental and closer to a theory. Theory is a jargon among Theoretical Messiahs for the longer version “equation of motion“. When two kinds of equation of motion, one for the wave and one for the particle are united into one, a feat called as unification we obtain equations whose specific example is Schrodinger Equation also called a wave equation of motion, with the implicit understanding that, such are conditions for particles and waves both, rather than just one, either of them.