Conduction is a property of the metallic bond in solids, and it's basically a process that can only be understood using quantum physics. We will discuss it in a crude classical framework.

The "conventional current" points in the same direction as the electric field E in the conductor.

A crude classical approximation to the behavior of conduction electrons in a metal with an electric field present.  Here vd is the average drift speed of a charge carrier through the conductor, due to the electric field.

In a conductor, applied voltage and current are proportional. This is not a "law," it's just a property of conductors, and serves as a definition of the concept of resistance.  V = R I, and R = ρℓ/A, with R the resistance and ρ as the resistivity, to factor out the dependence on the physical dimensions of the conductor. It is also convenient to use a vector quantity to represent current. This quantity is called the current density, j. It is defined to point in the same direction as E. In terms of current, the magnitude of the current density can be defined as j = I/A, where A is the area of the conductor. Ohm's Rule can then be written as j = E/ρ.

In a battery, an ongoing chemical reaction produces a constant potential difference between two different conducting plates.

Batteries themselves have an internal resistance, call it r. Therefore if we put a battery across a single resistance R, you can see via conservation of energy that the total effective resistance is just r + R. We will have much more to say about this in the next chapter!

In a semiconductor, the resistance DECREASES as temperature increases, exactly backward as compared to conductors. The higher the temperature, the more electrons are lifted to the conduction band, in a semi-conductor. Conductivity σ = 1/ρ.