Definition:

One of the fundamental principles of Scientology is the existence of the thetan, which is our ultimate self. Trauma and strife built over many lifetimes have buried the self-knowledge of the thetan with engrams. Auditing, one of the basic practices of Scientology, is a process which helps the practitioner remove engrams, become self-conscious of the thetan, and free one's full potential thought this self-knowledge.

Thetan vs. Soul

The thetan is often equated with the concept of the soul in many Western religions. However, Scientologists emphasize that people do not have thetans as Christians say they have a soul. Instead, the Scientologist will say that we are thetans. Rather than being both body and soul. Scientologists see humans as thetans who happen to have bodies. With this understanding it is accepted that a fully functioning thetan, unencumbered by engrams or other limiting negativity, can operate independently of a physical body.
The thetan uses the mind and animates the body. It does this because it thinks it has to use a body in order to interact with the universe.

Becoming Clear

Beginning Scientologists work to gain the state of Clear. The clear state involves an understanding that one's identity is completely separate from his body. As such, you are ultimately responsible for your own situation, which can only be as good as you accept it to be.
A Clear is also free of the “reactive mind,” which is the process by which negative events form harmful and limiting engrams.

A major aspect of progressing toward Clear is confronting fears. Fear makes us avoid certain situations, meaning that we are allowing the object of our fear to be in control. A Clear understands that he is in control, and that problems can only be solved when fully confronted. The auditing process helps Scientologists understand what they still need to confront in their own lives.

Benefits of Clear

A person who has reached the Clear state is expected to show self-confidence, willpower, intelligence, good health, creativity, and self-control. This is not the gaining of ability so much as the revealing of it. The thetan has always had these attributes, but doubts, fears and engrams have been masking them.

Beyond Clear

After reaching Clear, Scientologists gain knowledge of being Operating Thetans, or OTs. Just as the pre-Clears worked to get control over the self, OTs work to gain control over their surroundings, allowing their thetans to spiritually reach out and interact with the universe without the need of a body. At present, no Scientologist has progressed beyond OT VIII, although higher levels are theorized.
The ultimate state is known as Cleared Theta Clear, which is a godlike state. Such a thetan could "move MEST [matter, energy, space and time] and control others from a distance, or create his own universe; a person who is able to create his own universe or, living in the MEST universe is able to create illusions perceivable by others at will, to handle MEST universe objects without mechanical means and to have and feel no need of bodies or even the MEST universe to keep himself and his friends interested in existence."

Gnostic Comparisons

The Scientological view of the thetan is similar to that of the soul in Gnostic thought. Thetans are, at their heart, incredibly pure and powerful forces, but they have become burdened with the weight of the material world, which Scientologists work to confront and escape.
Gnostics see the soul as a divine piece of the godhead which has descended and become trapped in matter. Gnostics attempt to purify body and mind in order to one day escape the material world and ascend back to the godhead.

The one big difference is that Scientologists see thetans as individual beings, not as divine sparks of an even greater being.