"Begging the question" in a philosophical context means assuming what one is trying to prove. Typically it is a matter of having a premise that is very similar to one's conclusion. It does not make an argument deductively invalid, but means that the argument can be only trivially valid. For example:
- Someone asks, "You really claim to be the tennis player in the world?" I argue, "Sure, I'm the best player in the world at EVERY sport, so of course I'm the best at tennis."
This is actually a valid deductive argument, but the premise assumes the conclusion, and yet is itself controversial. Anyone who doesn't believe that I'm not the best at tennis is really not going to believe that I'm the best at every sport including tennis.
It is related to the ordinary usage because it offers a premise which invites or requires or begs an interlocutor to ask a further question: "but how do you know THAT?" or "why should I believe THAT?" Deductive arguments are practically good when they not only are deductively valid and have true premises but also are useful for persuading someone of something. An argument that begs the question is not generally going to persuade anyone of anything they don't already believe. It could only do so in very specific and unsual contexts (like if you didn't understand that tennis is a sport).
Mozibur Ullah's answer is very close. It's right that it's a matter of there being a premise that is neither self-evident nor argued for, and which therefore requires a further argument. But it's worth pointing out that begging the question is not a formal fallacy, meaning that it does not make arguments invalid (and it isn't typical to describe sentences like conclusions as valid or invalid).