Beg the question
This is one of those rare phrases in which the meaning is more debated than the origin.
The usage that has become common in recent years has a meaning something along the lines of 'prompt/raise the question', that is, 'beg that the question be asked'. This is usually seen in circumstances where something is described and then an explanation is sought; for example, this piece from a 2003 edition of the Jamaican newspaper The Gleaner:
What we are saying here is that every 2 days a juvenile is arrested and it begs the question, "What is really happening to our parents?"
Beg the question is now so widely misused by almost everyone that it has gone past the point of correcting them.
This usage is understandable and has presumably come about by interpreting the 'beg' of 'beg the question' as 'request' or 'humbly submit'. This is the meaning of the word in the similar phrase 'beg to differ'.
The original meaning was quite different though. To 'beg the question' was coined as a rather over-literal translation of the Latin phrase 'petitio principii'. The Latin version was itself a translation of Greek text 'en archei aiteisthai' taken from Aristotle's Prior Analytics. The phrase was known in English by at least 1581, at which date it was recorded by William Clarke:
"Ffiij, I say this is still to begge the question."
The logical constructs that Aristotle was describing were statements that assume the truth one is attempting to prove. Those might be questions which have an assertion smuggled into them, like 'Why has England fewer trees per acre than any other country in Europe?'. More commonly such statements contain the fallacious reasoning that we now usually call a 'circular argument'; for example, 'He must be speaking the truth because he never lies'. The 'truth' being assumed in advance isn't always so blatant. René Descartes' famous 'I think, therefore I am' can be said to be begging the question as he must exist before he can think - it is hardly a proof of anything to state 'I exist, therefore I am'.
If things weren't obscure enough with this phrase there was a version of the meaning that emerged between the two given above. That was its use to mean 'avoid the question'. This presumably also came from a misreading of 'beg' to follow the meaning of 'beggar description' or 'beggar belief'. That meaning of 'beggar', which seems to have been coined by Shakespeare in Anthony and Cleopatra, 1606, is 'exceed the resources of; go beyond':
"For her owne person It beggerd all discription."
Most authorities now view the current 'raise the question' meaning as acceptable, even if that is a somewhat grudging recognition that the weight of numbers of those who use it that way is overwhelming. It is also suggested by some that the minority who know and understand the original version should avoid using it, unless they are amongst consenting adults, as they aren't likely to be understood. That would be an unfortunate route to take. Whatever we might prefer, it is very likely that the percentage of the population that knows, or cares, that they are using the phrase incorrectly will continue to decline.