Electron holes in metals For semiconductors, conductivity is given by
$$\sigma=n|e|\mu_e+p|e|\mu_h$$
where $\sigma$ is conductivity, $n$ is the concentration of electrons, $e$ is the elementary charge, $p$ is the concentration of holes, $\mu_e$ is the mobility of electrons, and $\mu_h$ is the mobility of holes.
I do not understand why you have to include the holes for semiconductors and not for conductors. Yes, I have seen in other places that metals do not have holes because their valence and conduction bonds are usually indistinct from each other, whereas holes are left in the valence band when electrons move into the conduction band in semiconductors. But what if I don't want to think of conduction in terms of band theory, but classically?
In the sea of electrons model, an area that is vacated by an electron acquires a positive charge relative to its previous, electron-filled state. Thus, as an electron moves through a conductive wire, it is as though a positive hole is moving through the other way, and so we could consider current as being a flow of positive charges instead of a negative one. Serway's College Physics provides this diagram:

Thus, I fail to distinguish between the presence of holes in conductors and semiconductors. Whenever you have an electron moving, a hole is created in its absence, regardless of what that material is. So why is that you have to consider holes in semiconductors and not in metals?
 A: Atoms are really distinguished from each other due to their nuclear properties . It is the number of protons in the nucleus that defines the electric potential which will trap electrons and create a neutral atom. Thus metals have a specific type of nucleus that generates the potential which gives rise to loosely bound electrons in the outer shells. It is a complicated interplay between the potentials of electrons and protons that generates the electron shells described by the quantum numbers.
So it is a many body interaction that will keep electrons mobile in metals, and the positive "holes" immobile at the location of each atom.

In a conductor, electric current can flow freely, in an insulator it cannot. Metals such as copper typify conductors, while most non-metallic solids are said to be good insulators, having extremely high resistance to the flow of charge through them. "Conductor" implies that the outer electrons of the atoms are loosely bound and free to move through the material. Most atoms hold on to their electrons tightly and are insulators. In copper, the valence electrons are essentially free and strongly repel each other. Any external influence which moves one of them will cause a repulsion of other electrons which propagates, "domino fashion" through the conductor.

In semiconducting materials there is small mobility of both electrons and the holes left behind, again because of the potential solutions of the many body problem:

A silicon crystal is different from an insulator because at any temperature above absolute zero temperature, there is a finite probability that an electron in the lattice will be knocked loose from its position, leaving behind an electron deficiency called a "hole".
If a voltage is applied, then both the electron and the hole can contribute to a small current flow.
The conductivity of a semiconductor can be modeled in terms of the band theory of solids.

You cannot explain the behavior of matter below the nano scale , classically. Atoms and electrons are already quantum mechanical entities. It is the reason why quantum mechanics was invented and believed to be the underlying level of nature. Quantum numbers make a huge difference in the behavior of "particles" in the microcosm of atoms and molecules.
Edit looking at your figure: It is not talking of motion of physical positive charges, holes, but of hypothetical ones. In a semiconductor, the holes move, i.e.a neutral atoms become positively charged sequentially. In metals , the positive charge is attached to the individual atom, generating the potential that gives the energy levels. For holes to move, atoms at the atomic level have to exchange electrons. In metals the electrons are shared by all atoms and the mobility of holes is zero
The word "hole" is used in semiconductors as the subtraction of an electron from a neutral atom. The word "hole" that you use classically is the "space left from the motion of an electrin in a band"  Electrons in bands are associated with the whole crystal, not with individual atoms.
A: 
When we consider the conductivity of semiconductors, we consider $n_e$ i.e.number of free electrons,$i_e$ (current due to free electrons) and $n_h$ i.e.number of holes,$i_h$ (current due to holes). Now when we consider the net current, we need to take into account the current due to free electrons and the current due to bound electrons which is equivalent to the current due to holes as shown in the figure.
So, $$i_n=i_e+i_b=i_e+i_h$$
where $i_n$ is the net current and $i_b$ is the current due to bound electrons.

In conductors we consider only the motion of free electrons flowing. This is because the mobility of bound electrons (or holes) is very less and there are many free electrons (although equal to the number of holes or else wire would not be neutral) whose mobility is very much higher than the holes. Due to this contribution of holes is very small compared to the current due to electrons(i.e. very high number of electron pass through the cross-section of the conductor as compared to the number of holes). That's why we do not consider holes in a conductor.
But we have to consider both in semiconductors because there are comparable number of electrons and holes due to this. $i_b$ or $i_h$  cannot be ignored in comparison with $i_e$.
A: In semiconductors, the electrons that conduct currents are near the band edge. There, the dispersion law is usually parabolic (dispersion law is the relationship between particle energy and momentum). That is the same as one for a classical particle, which allows to think of electrons and holes as classical charged particles.
In metals, the electrons that conduct are somewhere in the middle of a band. The dispersion law is not parabolic there. So, the electrons behave in a much more weird ways, and one cannot treat them as classical particles. Attempts to separate them into electrons and holes make no sense. For example, if you use hall effect to determine what is the sign of carriers, you'll end up with different answers for different metals. it.stlawu.edu/~koon/HallTable.html
