Some years ago I was listening to a sports call-in radio show in Pittsburgh when the host, a particularly obnoxious know-it-all fond of putting down his listeners with his ten-dollar vocabulary, pontificated that the previous day's blown save by a Pirate reliever begged the question: why hadn't the manager left in the starter who after all had been pitching very well? This was the first time that I had ever heard the phrase, "begging the question," used in this sense; the first time I heard it used to mean, in effect, to raise the question. As far as I was concerned, to beg the question meant to avoid the question, to make a point that had nothing to do with the question.
So, for example, if someone had suggested that had the Pirates gotten more runs, the lead wouldn't have been blown, that might well be true, but it begs the question. The Pirates had the lead. The relief pitcher blew the lead. The idea that more runs might have helped is simply beside the point; it begs the question. Secure in my knowledge of the meaning of the phrase, I smirked and felt smugly superior to the blowhard radio voice, and went my merry way.
Turns out we both were wrong.
Since that first sighting, the use of the phrase to mean raising the question has grown like a viral video. On inventorpot.com, Steve Levenstein talking about a female robot invented by Tomotaka Takahashi says: "Takahashi believes that 'half of all robots will be "female" in the near future,' which begs the question... top half or bottom half?" Blogcomposters.com asks with regard to the Eco Pen: "Now, this begs to question, is a biodegradable pen that costs $2 preferable to the old fashioned Bics that run about 9¢ a piece?" I could go on, but you get the idea. Everybody and his brother use it in this sense, and fewer and fewer users, if any, seem to be aware of my own "more accurate," "correct" explanation of its meaning. Google the phrase and it quickly becomes clear that while neither I nor the talk show host have the most acceptable meaning, he may well have the more common meaning as used today, while I may have a definition somewhat closer to the more traditional one, if not quite on the button.
Traditionally begging the question refers to the logical fallacy of petitio principii, sometimes called circular argument. This basic sense of the term goes back to Aristotle. In a formal context, such as debate, it occurs when the conclusion is one of the premises of the argument. Thus for example this passage from Richard Whately quoted on Wikipedia: "To allow every man an unbounded freedom of speech must always be, on the whole advantageous to the State, for it is highly conducive to the interests of the community that each individual should enjoy a liberty perfectly unlimited of expressing his sentiments." In effect, what is being said here is that freedom of speech is good for the state because freedom of speech is good for the state. Examples can get more complicated, but this is the general idea.