To beg a question means to assume the conclusion of an argument—a type of circular reasoning. This is an informal fallacy, in which an arguer includes the conclusion to be proven within a premise of the argument, often in an indirect way such that its presence within the premise is hidden or at least not easily apparent.
The term "begging the question", as this is usually phrased, originated in the 16th century as a mistranslation of the Latin petitio principii, which actually translates as "assuming the initial point". In modern vernacular usage, "to beg the question" is frequently used to mean "to invite the question" (as in "This begs the question of whether...") or "to dodge a question". Many consider these usages incorrect in contexts that demand strict adherence to the technical definition.
The original phrase used by Aristotle from which begging the question descends is: τὸ ἐξ ἀρχῆς (or sometimes ἐν ἀρχῇ) αἰτεῖν, asking for the initial thing. Aristotle's intended meaning is closely tied to the type of dialectical argument he discusses in his Topics, book VIII: a formalized debate in which the defending party asserts a thesis that the attacking party must attempt to refute by asking yes-or-no questions and deducing some inconsistency between the responses and the original thesis.
In this stylized form of debate, the proposition that the answerer undertakes to defend is called "the initial thing" (τὸ ἐξ ἀρχῆς, τὸ ἐν ἀρχῇ) and one of the rules of the debate is that the questioner cannot simply ask for it (that would be trivial and uninteresting). Aristotle discusses this in Sophistical Refutations and in Prior Analytics book II, (64b, 34–65a 9, for circular reasoning see 57b, 18–59b, 1).
The stylized dialectical exchanges Aristotle discusses in the Topics included rules for scoring the debate, and one important issue was precisely the matter of asking for the initial thing—which included not just making the actual thesis adopted by the answerer into a question, but also making a question out of a sentence that was too close to that thesis (for example, PA II 16).