For my understanding:
Maxwell's equations combined with Poynting's theorem give us a model where electricity is energy carried along with the electromagnetic field (energy is stored in the field, not into kinetic energy of the electron in this model).
In around year 1900, the Drude model was created using kinetic gas theory and Boltzmann statistics. In this theory, electricity is carried by movement of "free electrons" that are accelerated by electric field and then "bounce" along the way. This is model I was taught in high school. We are able to get "Drift velocity" of the electron this way.
The theory was improved in 1933, combining the concepts of quantum mechanics and Fermi-Dirac statistics, resulting in Sommerfeld model that is still used in solid-state physics. Drift velocity is replaced with fermi velocity, and electrons that have fermi energy will take part in electric conductivity. With the quasi-particle model, semi conductors are explained pretty well and it's a very useful way to do it because even a single small grain of sand contains so many electrons that solving the exact Schrödinger partial differential equation would be practically impossible. Sometimes the real dynamics can be replaced by a quasi-particle model, which is based on the overall result of the interactions of a complex system.
So I'm now little bit confused about whole electricity. One model says we have some "moving electrons" on the wire/conductor, other says that we have probability for wave function to interact with neighbour wave function via photon field (or with a wave function more further away, the probability for this to happen would be smaller though), so the conductor/wire is just kind of tightly packed possibilities for electromagnetic interaction to happen by one way to look at it.
So my Question: Can someone explain to me what electricity is and how it transports on the conductors? Basis of the Drude model seem so different to Maxwell's model.
I don't know how quantum field theory works so I will just leave it out for now. When I look at some QFT equations they just looks like algebraic mess for me... well defined mess, but mess anyway.