Mark Liberman discusses 'beg the question' on Language Log. A commenter chips in.
As to whether 'begs the question' is an archaic term for all but a few specialists, I didn't grow up surrounded by formal logicians or professional philosophers but I've used it unselfconsciously since my childhood (OK, we're a pretty argumentative family…).
[(myl) As someone who uses the this phrase in its medieval-logic version without being a philosopher or an intellectual historian, you fall into the category that I described as "the few pedants". The "few" part of this description is not open to question, I think — the phrase is fairly common in the mass media, and roughly 99% of its uses are the kind that make you "seethe in silence", with most of the rest being meta-discussions in writings about usage. And "pedant" seems to me like a plausible description for people who maintain an irrational residue of high culture that almost no one else knows about. If it makes you feel any better, I'm obviously a member of the same category.]
This is a bit unnerving - I had no idea 'beg the question' was commonly used to mean 'raise the question', and can't remember ever having heard it used that way. (I did, admittedly, study philosophy, so what I now find to be the minority usage seems natural.) ML, anyway, offers an excellent discussion of the history of the phrase.
[Er. I see Languagehat thinks the commenter is cut off from reality and we should dispense with this silly phrase. A slight problem is that I frequently find myself in arguments with people who assume the conclusion. I naturally want to point out that the argument is flawed, but the use of formal, academic language is often perceived as cold, aggressive, hostile; an 'ordinary language' phrase always seemed ideal for the situation. The point being, all this time I was under the happy illusion that ordinary language actually provided a phrase that was fit for purpose. The thing that's revealing about the discussion is not that the phrase is more esoteric than I had realised, but that the concept turns out, as far as I can see, to have no place in the culture at large.]