The phrase "begging the question" is supposed to mean assuming what you are trying to prove (and it still means this in logic). An argument that begs the question has an assumption that hasn't been proven and therefore can't be assumed; this unproven assumption leads naturally to the conclusion, which is also unproven. More conversationally, it means to assume something not yet proven or done: "Shopping now for a wedding dress is really begging the question--she hasn't been proposed to yet."
Unfortunately, "begs the question" is now generally used to mean "raises the question"; it is used this way so often that a careful writer can no longer use it correctly. Which is too bad, because begging the question is still a useful concept that deserves a phrase - especially during the presidential elections. Luckily, it is very close to the concept of circular reasoning, in which two related assumptions are used to prove each other (simplistically: "The Bible says God exists. The Bible is the word of God, so it must be true. Therefore God exists."), so there is a useful term still available to unravel political arguments.
Come to think of it, it is really hard to find a good example of an argument that begs the question, so maybe it isn't too bad a phrase to lose. Now if journalists would just quit using it the wrong way.