If today were April 30 (actually it is), what does 'later this month' mean?
If today were April 15, what does 'later this month' mean?
If today were April 1, what does 'later this month' mean?
ADD:
If today is Monday, when I say 'this Friday', I would mean the coming Friday. So, I thought 'this' might have some meaning relevant to some future.
For question 1, I thought the answer could be later half of May, couldn't it?
1) Possibly a sarcasm aimed at someone who's unaware of today's date.
2) Any day between 16th of April and 30th of April
3) Any day between 2nd of April and 30th of April
However, that being said, I suspect this is not what you are really asking. You will need to give more context as to when and where each of these conditions is being used/mentioned.
Update:
For question 1, I thought the answer could be later half of May, couldn't it?
No, that would be logically wrong. You'll have to say later next month or a similar expression to that effect.
If today is Monday, when I say 'this Friday', I would mean the coming Friday. So, I thought 'this' might have some meaning relevant to some future.
This opens a different can of worms which should be answered in another question. I always use this [day of the week] to refer to a day in the same week as the one I am in now to avoid confusion. Depending on the week order in your culture, this could be Sun - Sat or Mon - Sun. You might want to ask this in a different question, if you are confused about this, which I totally understand.
If someone told me on April 30th that she'd complete the project "later this month," I would presume she'd have it done by the end of May, not by the end of the day.
Sure, "within a month" would be a much more accurate way to phrase it, but I doubt I'd be confused over her meaning, unless this coworker was already known for such dry humor in her everyday speech, and I suspected she was hinting that the work was already just about complete.
The primary meaning of month regards a calendar month, but a secondary meaning is "a period of about four weeks." Many monthly cycles don't start on the first of the month, such as billing cycles and lunar cycles.
In writing, I'd fix it; that is, I'd change "later this month" to "within a month." Conversationally, though, I don't think I'd be thrown.
That said, in statement #2 ("later this month," spoken on the 15th), I'd probably assume that the word this signified we are talking about a period of about two weeks, not four – that is, until the end of the calendar month – but there could be some exceptions (such as if I was talking to my credit card company, whose months always seem to start and end on the 10th of every calendar month).
Most of the potential for ambiguity in these examples is found in the word this, as James pointed out above. We encounter the same potential for ambiguity when we say "later this week," or "later this year."
