Dynamically, more natural notion is phase space, which is the space where the evolution is first order in time. Intuitively, it's all the information you need to make the system evolve. Configuration space then arises when phase space has additional structure. Typically, when it is the cotangent bundle of a manifold, configuration space is naturally identified as the manifold in question. In a lot of cases, it makes sense to view configuration space as having half the dimensions of phase space. Intuitively, this works if you can define a Lagrangian/Hamiltonian, or more abstractly a symplectic structure (the fact that your phase space has even dimensions is already a good indicator). You cannot define configuration space without considering the dynamics as well.
For electromagnetism in vacuum, there are two degrees of freedom ($2$ polarisation states). And there are many ways to see this.
You can get this directly from <PERSON>'s equations. 40
You start with $6$ components of the EM fields. You get rid of $2$ due to the "static" <PERSON> equations: $$\nabla \cdot E = 0\ \nabla \cdot B = 0$$ so you have $4$ components. Since the equations are of first order in time derivative, this means that $4$ actually corresponds to the dimension of phase space, so halving it, you obtain the $2$ dof's.
From the potentials, you can get the same results. The idea is that <PERSON>'s equations can be derived from the Lagrangian density: $$ \mathcal L = \frac{1}{2}(E^2-B^2) \ = \frac{1}{2}\left((\nabla V+\partial_t A)^2-(\nabla\times A)^2\right) \ $$ This time, it looks as if there are $4$ dof's. However, the time derivative of $V$ does not appear, so it is not a dynamical variable, but rather enforces a static constraint (a Lagrange multiplier). So you only have three dof's. Finally, adding gauge invariance, you loose and additional dof, so you obtain $2$ as well. Note that you can similarly get the same result by specifying the gauge (<PERSON>, <PERSON>...).
If you are familiar with general relativity, you can similarly prove that the <PERSON>'s equations actually leave $2$ dof's even if you seem to have $10$ dof's for the metric tensor (with the added subtlety that there are more constraint equations and more gauge transformations).
Hope this helps. 418