Learn English – What does the phrase “Begging the question” mean

It should mean avoiding the question (or “to improperly take for granted”), when used in the original sense:

a type of logical fallacy in which the proposition to be proved is assumed implicitly or explicitly in the premise.

Begs the question:
(petitio principii, "assuming the initial point")

An argument that improperly assumes as true the very point the speaker is trying to argue for is said in formal logic to “beg the question.” Here is an example of a question-begging argument:

“This painting is trash because it is obviously worthless.”

The speaker is simply asserting the worthlessness of the work, not presenting any evidence to demonstrate that this is in fact the case

Most people now suppose the phrase implies something quite different: that the argument demands that a question about it be asked—raises the question. Although using the expression in its original sense is now rare, using it in the newer sense will cause irritation among traditionalists.

More recently, "to beg the question" has been used as a synonym for "to raise the question": for example,

"This year's budget deficit is half a trillion dollars, 
 which begs the question: how are we going to balance the budget?"

For a complete reference on the topic: fallacyfiles.

As Benjol mentions in the comments Eric Lipert has blogged about this:

When I say "begs the question", I mean it in the traditional sense of "this argument is fallacious because it takes as a premise an assumption which is at least as strong as the thing being proven, and is therefore an unwarranted assumption."

This pseudo-explanation has no predictive power; it doesn't tell us anything new, it just circles back on itself. The explanatory assumption is stronger than the thing we are trying to investigate.

Question-begging is not the act of raising more questions. Every good explanation raises more questions.
What makes this explanation a good one is that it is testable and has predictive power.

If you ask "why is this code thread-safe?" and the answer is "because it can be correctly called on multiple threads", we've begged the question. Why is it thread-safe? Because it's correct.
Why is it correct? Because it's thread-safe. Again, we have learned nothing about the nature of thread safety.

Phil Koop commented on that blog post:

The traditional meaning of "begging the question" is derived from an antiquated usage of "beg" meaning "to take for granted." In that usage, a "beggar" was usually someone who camped on the local squire's land without permission.

When this phrase had currency, it was in the dialect of people who had an enviable station in life; they were wealthy and powerful.
The examples of its usage that survive today have been preserved because they were uttered by people who had something interesting to say and were good at saying it, but it is also true that people paid attention to what they said because of their wealth and power.

So why do people now say "beg the question" when they mean "raise the question"?
As Ben says, it is quite an unnatural construction in the latter usage.
In my opinion, the attraction is that it imitates the language of those who once did use it naturally, and the modern speakers hope that the virtues of these older speakers - intelligence, fluency, and social status - will somehow be transferred to them.
It is just an unfortunate irony that the very misuse of the phrase should undermine the hopes that gave utterance to it.

Although it is true that language changes continually, this observation by itself is not an adequate guide to effective usage.
If I wanted to communicate with the widest possible audience, I would avoid the phrase altogether.
But in a blog for educated professionals, I see nothing wrong with promoting familiarity with the interesting ideas of the past.