Bells' definition of begging the question is correct. But whenever you see a journalist or hear a sports- or news-caster use this term, e will be using it in a different sense. 98% of the time, the mass media (in the USA) uses 'begs the question' to mean 'raises the question.' (About 2% of the time they will use it to mean 'avoid the question').

These people probably aren't all that bright (although every last one of them has had a college education), and they probably have no idea that begging the question is a logical fallacy. But on the other hand, many of the people who come to this node for information may be wanting to know what these TV personalities are talking about. Like it or not, the popular usage should be defined. And then eradicated.

I personally have only heard Begging the Question used, in philosophical context, to mean that the argument assumes what it is trying to prove.

Using 'beg the question' to mean 'assuming something that needs to be proven' is a somewhat archaic usage, although it still pops up from time to time. This older definition is arguably redundant, because you can always say that any or all premises in any argument need to be proven. Philosophers have a lot of fun questioning Time, Space, Material Objects, Personal Identity, Good, Bad, and anything else that stumbles into their path. If you find something that doesn't need to be proven, you will go down in history (along with Descartes -- but we're still arguing about whether or not 'I think, therefore I am' really works.)

As far as the form of an argument goes, it can be valid even while containing misinformation, and it can never be sound if you require a perfectly rigorous proof for everything. But an argument is not useful if it assumes what it is trying to prove, therefor Begging the Question is an informal fallacy*.

A simple example of begging the question might be something like this:

1) I am the smartest person in the world.
2) How do we know this? Well, it must be true because I say so.
3) Why should you trust me? Well, because I'm the smartest person in the world!

Of course, most cases of begging the question are much more subtle (and less silly) than that. Take a look at this example.

1) God, by definition, is a necessary being
2) By definition, a necessary being exists
3) Therefore, God exists.

Should be
1) If God exists, he is, by definition, a necessary being.
2) By definition, a necessary being exists
3) Therefore, if God exists, he exists.

Begging the question is a too-literal traslation of the original Greek phrase en archei aiteisthai, meaning something like 'in the beginning to ask'. Aiteisthai ('ask') can be translated as 'beg', and back in the 1500s it was translated this way. In modern usage this sounds odd, but in light of the fact that it has been used for 500 years, I strongly support the traditional usage of this phrase. After all, if you mean to say 'raises the question', then why not just say it?

If you really don't like the phrase 'beg the question', this fallacy is also called Petitio Principii, Circulus in demonstrando, circular argument, circular reasoning, circulus in probando, or even sometimes a vicious circle.

* I had originally claimed that begging the question invalidated an argument; skeller has pointed out that an argument can be valid even if it begs the question:

Therefore X

This is an entirely correct argument form, even if you have no reason to believe that X is true. The problem with begging the question is that it doesn't give you sufficient evidence to believe the conclusion, not that the argument form is incorrect.