I am studying Electrodynamics and I have been introduced to the concept of Gauge Invariance.

This was introduced by noting that $E$ and $B$ amount to 6 six degrees of freedom and the Maxwell equations amount of 3 degrees of freedom. On the other hand, if we write $$E = - \nabla \phi - \frac{\partial A}{\partial t}, B = \nabla \times A$$

this contains 4 degrees of freedom.

The extra degree of freedom forms part of this Gauge invariance.

My lecture notes go on to talk about the Neumann gauge and the Lorenz gauge and how these are both 'natural' choices for a Gauge.

I have come to Stack Exchange because I am fairly confused. I am not sure what a 'gauge' even is and what their point is. It's not obvious from what I've put above...

Furthermore, I read on in my lecture notes and it says that in the Lorenz gauge, $A$ and $\phi$ satisfy wave equations. Again, I don't see how this is useful, but maybe a user on here can shed some light on gauges and this will make sense.

I am studying Electrodynamics and I have been introduced to the concept of Gauge Invariance.

This was introduced by noting that $E$ and $B$ amount to 6 six degrees of freedom and the Maxwell equations amount of 3 degrees of freedom. On the other hand, if we write $$E = - \nabla \phi - \frac{\partial A}{\partial t}, \qquad B = \nabla \times A$$

this contains 4 degrees of freedom.

The extra degree of freedom forms part of this Gauge invariance.

My lecture notes go on to talk about the Neumann gauge and the Lorenz gauge and how these are both 'natural' choices for a Gauge.

I have come to Stack Exchange because I am fairly confused. I am not sure what a 'gauge' even is and what their point is. It's not obvious from what I've put above...

Furthermore, I read on in my lecture notes and it says that in the Lorenz gauge, $A$ and $\phi$ satisfy wave equations. Again, I don't see how this is useful, but maybe a user on here can shed some light on gauges and this will make sense.