The actual story about "beg the question" is far more complicated than virtually all authorities suppose. In its logician's use: it is always used absolutely, that is, "That just begs the question" (full stop: there's no further "question"). In that sense, it means "Assume the very thing you're trying to prove."
That is, of course, a bizarre construction, and it is only explainable on the basis of its historical origin, which descends from a phrase in Aristotle's logical works that can't really be understood without knowing the context in which it arose: a dialectical argument as portrayed in Book VIII of his Topics. As depicted there, a dialectical debate has two participants, a questioner and an answerer; the answerer begins the debate by undertaking to defend some proposition, and the questioner then tries to refute the answerer by asking questions that can be answered by yes or no of the answerer and, from these, deducing the contradictory of the answerer's initial proposition. In this context, Aristotle calls the initial proposition proposed by the answerer "the initial thing" (to en archêi).
One of the rules of this kind of debate is that the questioner cannot simply make the very thesis put forward by the answerer into a question and ask that: to do so is the error of "asking for the initial thing" (to to en archêi aiteisthai). It also counts as "asking for the initial thing" if the questioner asks something not identical to but more or less equivalent to the initial thing, though explaining how close it has to be is a problem for Aristotle.
Translated out of the context of dialectical debate, this same error can be described as "taking the conclusion you are trying to prove as one of the premises of your argument", and that is the original sense of "begging the question" when it enters English in the late 16th century ("question" having taken on the sense "the proposition being debated").
A complicating point in this history is that the Greek verb aitein, "ask", took on the sense "ask for as a premise" in logical and mathematical contexts and thus came to mean "assume" (as a premise). If we assume the same for "beg" in English, then "beg the question" can indeed be read as "assume the thing you're trying to prove", although that is hardly what an ordinary speaker of English would suppose it to mean.
(As a footnote: the translation "in the beginning to assume", given in some authorities as a translation of to en archêi, is grammatically impossible as a rendering of it in Aristotle, since it assumes the definite article to goes with the infinitive aiteisthai; Aristotle's usage, especially in those contexts in which the phrase begins with two definite articles, shows that it goes instead with "in the begining" (so, "to assume the in-the-beginning thing" would be much better).)
begging the question
has any relation to the meanings of the wordsto beg
anda question
. Of course I was interested in the origins of the phrase, too -- anything that I couldn't find answered elsewhere. Nicholas and Peter Shor did a good job at answering already, thanks!