The phrase "beg the question" can be used to mean two very different things. In its original usage, it implied that asking the question implied or assumed the answer to another question. "Beg the question" is also used to describe a situation or circumstance that begs for a specific question to be asked.
Aristotle's writings about logic seem to be the origins of the first meaning of the phrase. In the English translation of his works, "beg the question" was used to refer to a corollary question that assumes the answer to the first question. Specifically, it means that the asker is assuming that she knows the correct answer to the initial, unasked question. An example of a situation that might beg the question would be a man who asks a woman when he can pick her up for a date without first asking if she will go on a date with him. The time question assumes that if he did ask her to go out with him, the answer would be yes and that, therefore, he can skip straight to asking about the time.
Begging the question is a technique often employed by sales people attempting to convert a prospect to a customer. For example, a sales clerk in a store might ask a customer whether she wants to the blue sweater or the green sweater. The clerk is implying that the decision to purchase a sweater has been made and that the customer must now merely decide which she wants. When this technique works, it works well. When it doesn't work, however, the customer is likely to become mistrustful of the sales person, a situation that may cause the sale to be lost.
Sometime in the early 1990s the phrase "beg the question" took on a secondary meaning. It began to be used to describe a circumstance that compels one to ask a specific question. For example, if someone states that his car blew up, it would be almost impossible not to ask how it happened.
This usage can be used to lend a question an air of seriousness or urgency. For example, in the face of a heinous crime, an impassioned editorial might state that the crime begs the question of how it could have occurred. In some cases, the phrase may simply be substituted for the phrase "asks the question," usually to imply a greater level of gravity or desperation.