What is Molecular Orbital (MO) theory?

Molecular orbital theory is an expansion on VSEPR theory (check the link if you need a referesher), which you should've learned about in High School. MO theory differs from VSEPR theory, in that it takes into account regions of space spread over the entire molecule in which you might find an electron. MO theory allows us to calculate these areas, called molecular orbitals, the nuclei of the atoms are placed in their equilibrium position for these calculations. Molecular orbitals form from interactions arising in the molecule, between the nuclei and the orbitals surrounding them. These interactions are restricted by several factors:

Interactions are allowed only if the symmetries of the atomic orbitals are compatible, i.e. they can 'fit' together;
The relative energies of the atomic orbitals is also very important, as the further apart in terms of energy each atomic orbital is, the less likely they will be to form a molecular orbital;
the region of overlap between atomic orbitals needs to be significant enough for a MO to form.

The amount of MOs that form always equals the amount of atomic orbitals present in the molecule. It's important to remember that all MOs have associated energies, so electrons will fill MOs according to the Aufbau principle. You can sum all the energies of the MOs to get the total energy of the molecule.
So what if we apply this so a very simple molecule, such as H₂?

MO Theory in H₂

Molecular orbitals are indicated by wavefunctions, 𝜓, which describe the nature of electrons within that orbital. Since electrons are also considered waves, the overall sign of the wavefunction can either be + or -. Additionally, just like waves, they can be described as transverse waves, i.e. they can interferre in a constructive (in-phase), described as "bonding" and destructive (out of phase), described as "anti-bonding" in MOs. We can describe these types of bonding orbitals as linear combinations of their respective atomic orbitals. The respective mathematical functions of these orbitals can be simplified as 𝜓MO(in-phase) and 𝜓*MO(out-of-phase), where N is a normalizing constant:

𝜓_MO(in-phase) = 𝜓_MO = N[𝜓₁ + 𝜓₂]

𝜓_MO(out-of-phase) = 𝜓*_MO = N[𝜓₁ - 𝜓₂]

N can be determined the equation below, whee S is the "overlap integral", a measure of how the regions of space:

N = 1 / √2(1+S) ≈ 1/√2

N* = 1 / √2(1-S) ≈ 1/√2

S is not taken into account in the approximate values given. The diagram above shows that the bonding MO, 𝜓_MO, is stabilized by the Is orbitals from the contributing H atoms, to form the bonding orbital. Conversely, the anti-bonding MO is destabilized, because the difference in energy between the anti-bonding MO and the 1s orbitals is slightly bigger than that between the bonding MO - this effect is far to complicated for me to explain in my blogs, it is however important to remember.
We can represent the anti-bonding and bonding orbitals in 3 dimensions, as anti-bonding orbitals form nodes due to destructive interference; the first orbital shown is the highest occupied molecular orbital, or HOMO for short, and the second is the lowest unoccupied molecular orbital, or LUMO. The HOMO is the bonding orbital as and the LUMO is the anti-bonding orbital:

You would notice that there is separate colours in the LUMO, indicating that there is a node in between and the 1s orbitals that make up the MO are out of phase.

Bonding in other X₂ type molecules

Now let's have a look at other molecules, say F2 and O2, which have 2s and 2p orbitals; recall from VSEPR theory that these orbitals have a maximum of 2 and 6 electrons each, respectively. The bond order can be calculated like this:

Bond Order = 1/2(total bonding electrons - total anti-bonding electrons)

We are only concerned with the valence shells, so we'll only display it as the 2s and 2p orbitals. The number of MOs equals the amount of atomic orbitals (as in VESPR), from this we can determine that the HOMO is 1/2 number of total valence electrons, LUMO on the other hand is the HOMO+1. Using the Aufbau principle, we can draw up a diagram as with Hydrogen, so we get (remember this is not starting at the 1s orbital):

Using chemistry software it is possible to model these MOs using quantum calculations, various types of these calculations exist with higher or lesser degrees of accuracy. The HOMO can be seen below: