For example, "All cats are evil, otherwise you would not see cats do evil things."
The premise of this statement is that cats are evil, and the justification is that you see cats do evil things. The statement forms a circular argument.
But, there's another common use of the term that often appears as , "begs the question", as in, "your statement begs the question of who will do this work". It means that there is an obvious or ignored question that arises from a statement.
There are grammarians and logicians that will argue that this is somehow the wrong use of the term, such as the website begthequesiton.info (which dedicates itself to this topic). Ironically, these individuals often employ logical fallacies to disregard the modern usage of the term. There are people that seem gleefully unaware of how English works. Common usage is correct usage. Dictionaries now list the modern usage on equal weight as the logical fallacy definition. See idioms area on freedictionary.com.
There is often a claim that using the phrase in the modern sense is somehow confusing (see some of these claims on QuickAndDirty.com by Grammar Girl). However, common usage is so prevalent, there is no confusion as to when the term is being used one way or another. If someone wishes to distinguish between the logical fallacy and the assertion of an obvious/ignored question, then they do so with context, just as they would for the use of any other common terms with multiple meanings. This begs the question, why is denial of the validity of the modern definition so important to some people?