Here are three of the countless examples of beg the question one can find online: "It begs the question of who Fluke really is." "Exports' clout begs the question: Was NAFTA good or bad?" "He did stand-up comedy once, which begs the question, What can't this guy do?"
In each case, the writer should have said "raises the question" or "suggests the question" or "demands the question."
A succinct definition of beg the question is found in H.W. Fowler's Modern English Usage: "The fallacy of founding a conclusion on a basis that as much needs to be proved as the conclusion itself." Fowler offers this example: "Capital punishment is necessary because without it murders would increase." There are two unproven assertions in that sentence, and yet the second one is supposed to prove the first.
Here's another kind of question-begging: "Good grammar matters because proper speech or writing makes a difference." In this instance of begging the question, the "proof" is merely the premise restated in different words. That's like saying, "Good grammar matters because I just said so."