Covalent bonding is a form of intramolecular bonding that occurs as a result of a sharing of electrons between atoms, usually occuring between elements of similar electronegativity. Electrons alternate between two or more atoms, causing the atoms to attract. This bond imparts certain key characteristics to the resulting compound, such as polarity, differing bond numbers and lengths, differing bond enthalpy, and differing bond angles and three-dimensional shapes based on VSEPR theory.
First of all, polarity may result from a covalent bond. When one or more of a compound's elements has a higher electronegativity than the others, it will attract the bonded electron(s) more strongly, causing the electron to exist in the orbital of this atom more often than in the other atoms in the compound. As a result, a net negative pole is observed on highly electronegative elements in a compound. Of course, as the electron is more often in the more electronegative atom, the less electronegative atom contains the electron less often, and therefore becomes a positive pole. These poles become involved in intermolecular dipole-dipole interactions.
This form of bonding occurs based on the electronic configuration of the atom. Orbitals that are filled are more stable and less likely to bond than orbitals that are only partially filled. Orbitals that are partially filled tend to accept an electron from another atom with a partially filled orbital and form a bonded pair. The bonded pair is then shared between the atoms, contributing to the stability of each. Of course, filled orbitals can still participate in covalent bonding. This is known as a co-ordinate covalent bond. In this type of covalent bond, a pair of electrons originating from a single atom is shared.
Bonds need not consist of a single electron being shared between atoms. When two electrons are shared between two atoms, a double bond is formed. When three are shared between two atoms, a triple bond is formed. Double bonds are stronger than single bonds, resulting in a shorter bond length. Triple bonds are even stronger, and as a result are even shorter. As a result, it is extremely difficult to separate two atoms of nitrogen (which are triple-bonded) compared to separating two atoms of hydrogen (which are single-bonded).
In the previous example, nitrogen atoms were said to be more difficult to separate than hydrogen atoms. This is because nitrogen requires more energy in order to break the bonds between atoms. The amount of energy required to break an intramolecular bond is known as its bond enthalpy. The bond between two atoms of nitrogen has a high bond enthalpy, while the bond between two atoms of hydrogen has a low enthalpy.
The differing properties of covalent bonding culminate in VSEPR (Valence Shell Electron Pair Repulsion) theory. This theory is actually a model used to predict the three-dimensional shape of a molecule based on its Lewis Structure (a diagrammatic representation of the electronic configuration of the molecule.) The molecule can take on a number of basic shapes, based upon the number of atoms each atom is bonded to, as well as the locations of any un-bonded pairs of electrons in the molecule.
These are only the fundamentals of covalent bonding. This form of intramolecular bonding occurs in many other forms, each with their own set of interesting properties. I would recommend researching aromaticity and covalent network substances for more variations on covalent bonding.