Lines of circumvallation/contravallation are two different ways of arranging your forces during a siege.
Interior/exterior lines of communication are two different ways of maneuvering your forces.
Lines of circumvallation:
The military tactic of laying siege to an enemy force by arranging your forces in an arc which faces toward a besieged army. This is primarily done to protect from sorties by defenders and to enhance a blockade. Lines of circumvallation usually consist of earth ramparts and entrenchments which can be used as a base for launching assaults.
Lines of contravallation:
The process of an attacking army arranging its' forces around an enemy force in an arc which faces away from the besieged army. This is primarily done to protect from any counter-attacks by the allies of a besieged army and to enhance a blockade. Lines of contravallation also usually consist of earth ramparts and entrenchments and are built parallel to the lines of circumvallation (thus protecting an attacking army in a double line of fortifications).
By allowing troops to be more easily redirected from besieging operations to the defense of the army, the circumvallation and contravallation tactics allow an attacking force to benefit from a limited amount of interior lines of communication.
If a strong enough force comes to the relief of the besieged army, then the attacking force may find themselves trapped within their own lines.
Interior lines of communication:
These are the relatively short and secure routes within a defenders given arc of operations. These provide mobility advantages which are not available to enemy forces because - the interior distances of all points in an arc are much shorter than the exterior distances.
By being able to rapidly concentrate combat power at vulnerable or decisive points, the defender can shift forces more quickly to meet an enemy attack than the enemy can shift its forces to attack these areas. Such routes also allow the defense to more effectively sustain their forces with provisions and provide better over all command and control.
Risk of being completely encircled and destroyed.
Exterior lines of communication:
These are the relatively long and insecure routes within an attackers cone of operations. Generally speaking, if a defender possesses interior lines, then an attacker almost always operates on exterior lines.
An attacker using exterior lines can launch assaults simultaneously all along the defensive line, thereby preventing the defense from massing his forces. Also, if victorious, an attacker stands a good chance of completely encircling the defender and totally annihilating him.
Such routes not only make it difficult for military commanders to concentrate their forces rapidly at decisive points and sustain them after arrival, but may also be difficult to safeguard if they traverse hostile territory.