Petitio principii

Begging the question, which goes by the technical name petitio principii, is a logical fallacy that occurs when an argument implicitly assumes its conclusion in a premise.

Examples
Consider the following argument: Arranging this argument in the usual premise–conclusion sequence reveals the problem with it: The premise assumes that God exists in the first place, otherwise he couldn't "do" anything.
 * I know God exists because of all the good things He has done for me in my life.
 * 1) God has done good things for me in my life.
 * 2) Therefore God exists.

A more subtle example, perhaps, is the following: The problem here is the assumption that what one saw was indeed "paranormal" rather than, for example a trick of the light or a hallucination. Under most people's definition, ghosts are examples of paranormal phenomena. However, one must accept that paranormal phenomena are possible to accept the premise that a "ghost" was actually seen. Since everything we can see is made of ordinary matter — i.e., emits or reflects photons according to well understood laws of physics — the underlying assumption that the "ghost" was actually outside of the ordinary laws of physics — i.e., paranormal — cannot be accepted without question. Again, reformatting the argument (and rewording the parts slightly) clarifies the problem: Note that the structure of this argument is entirely valid. We can use it for any number of real phenomena (e.g., "I have seen the moon, so I believe in the moon"). Since examples of begging the question can occur in logically valid arguments, petitio principii is considered an informal fallacy. The above example is also based on an enthymeme (a missing premise), namely that the "ghost" was a paranormal phenomenon.
 * I believe in the paranormal because I've actually seen a ghost.
 * 1) I have seen a paranormal phenomenon (a ghost).
 * 2) Therefore I believe in paranormal phenomena.

Different meanings
Sometimes the phrase "begging the question" refers to using a premise that requires at least as much justification as the intended conclusion — that is, that the use of the premise to justify the conclusion is at least as problematic as simply asserting the conclusion without any premise at all. For example: In this case the premise, that Jesus was the son of God, requires at least as much justification as the claim that the teachings of Jesus should be followed. That is, the argument moves backwards, in a sense, to the more fundamental question, "How do you know Jesus was the son of God?"
 * The teachings of Jesus should be followed because he was the son of God.

People often refer to "begging the question" when they really simply mean "raising another question". For example:
 * "You say that theism makes more sense than atheism, but that just begs the question of which religion/philosophy I should choose to believe."

This is not an appropriate use of the term, since the question being raised is not implicit in the preceding statement.