One of the most exciting parts of studying chemistry is the power to predict how atoms will combine into different molecules.  Of course, in order to do this, one must understand how the individual atoms behave alone and in close proximity to other atoms.  Chapter 12 of the physical chemistry textbook is devoted to chemical bonding and its two main theories:  valence bond theory (VB) and molecular orbital theory (MO).
As usual, I like to take an overall approach before tackling specific details.  There are multiple theories used in chemistry when discussing bonding but they all fall into one of two approaches.  The first is the visual approach that primarily deals with how the molecule would look and its overall structure, the second is a more detailed approach that explains and or predicts a molecules properties and how and why it would form and react.  This second approach often provides mathematical proof.
Before bonding can ever be explored though, the properties of the individual atoms need to be understood.  This visual approach is covered by Lewis dot structures and VSEPR (Valence Shell Electron Pair Repulsion) theory.  Quantum mechanics, specifically atomic orbitals, discussed in chapter 11, cover the arrangement of the electrons to help predict properties and re activity.
Lewis dot structures use the valence electrons, ones in the outer most shell, to show how the electrons would be available to bond.  It is based on the idea that pairs of electrons are favorable either inside bonds or as lone pairs with areas with unpaired (single) electrons as potential bonding sites.

Bonding using the Lewis dot backbone describes two types of covalent bonds, single and multiple and includes lone pairs of electrons.  Single bonds are created by sharing two electrons between atoms and are always formed first by the direct overlap of atomic orbitals.
If more than two electrons can be shared in one location there is the possibility of what I call “interacting” of other close atomic orbitals to form a multiple bond.  The atomic orbitals used in multiple bonds do not overlap as effectively as in the single bond case resulting in a weaker bond.  Sharing four electrons is called a double bond and sharing six electrons is called a triple bond.
 Purpose:  Fairly simple but very effective bond predicting tool for beginning students that does not require knowledge of atomic orbitals (they are used, but they just don’t know it).
VSEPR theory uses the final electron arrangement, based on Lewis dot structures, to predict the shape around individual atoms inside a molecule.  It is centered around five main “families” or arrangements, which depend on the number of “things”, either lone pairs or bonds, around the center atom with the overall shape changing slightly depending on the number of each.  See interactive in section 12.4 (screen grab above) of the physical chemistry textbook
# of “things”                  Arrangement
1 or 2                           Linear
3                                  Trigonal Planar
4                                  Tetrahedral
5                                  Trigonal Pyramidal
6                                  Octahedral
Purpose:  Basic arrangement in 3D space, shape.
Atomic orbitals are more concerned with the actual location and energy of the electrons and can be written in electron configuration or a more visual orbital diagram.  While providing valuable information, atomic orbitals are not particularly helpful in predicting bonding.

Purpose:  Electron location and energy.
Now that we understand the theories used to help predict bonding, the more complex bonding theories in chapter 12 can be explored, which I will discuss in the future.