My brother was saying that is grammatically correct to say he went to sleep at one today because it was after 12 o' clock, but I say it should be he went to sleep at one o'clock yesterday. Which one of us is correct?
This is not a matter of grammar but of semantics and idiom. I don't think most native speakers would use either "today" or "yesterday"; we'd say
I went to bed at one o'clock last night or
I went to bed at one o'clock this morning.
Good question! I agree with the comment that choosing "last night" or "this morning" is a matter of emphasis - If you want to stress that it was inconvenient or unusual for you, I'd suggest using "this morning"; if you were out with friends or it's normal for you to go to sleep in the early AM hours, I'd suggest using "last night".
As a matter of personal convention, I use 5am as the cutoff between "last night" and "this morning". At 5am it's unambiguously the morning or a new day, but 2 or 3am could easily still be considered a continuation or part of the same night that technically ended after 11:59pm
rexkogitans's comment to your question is correct. What you're asking is not an issue with the sentence's structure, and is not a question about grammar.
Most people will say that the day starts at midnight (12:00am). Some religious groups have been known to say that a new day starts at sundown, which would make this question moot.
Presuming that you accept that the new day starts at midnight, the technically correct answer is that 1:00 am was from today. However, if the person normally starts sleeping somewhere from 8pm-11:30pm, and simply did an unusual thing by starting sleep later on this particular night, then some people might say "I didn't go to bed last night until 1 am". Even though this wasn't actually "last night", that is when the person started the sleep session that would typically have started last night, so people might decide to use words that act that way.
While such a choice of words would be considered technically inaccurate, I have certainly heard people talk that way. I side with your brother, but many people would understand what you mean, and there's a good chance that it wouldn't cause much confusion.
Let's look at this another way. Today is Wednesday (at least where I am!) and brother went to bed at 01h00 on this day - this morning. If you think that it's yesterday, surely he'd have gone to bed at 01h00 on Tuesday. But it wasn't: it was Wednesday, the same day it still is. So logically, the correct phraseology is 'I went to bed at 01h00 today'.Which is just one hour after today (Wednesday) started. Not 25 hours ago...
Day or night doesn't really come into the equation - you both might be at the North Pole for all we know!
Well if one considers morning after the sunrise - then saying yesterday night at one is correct but if the one is considering morning past 00 hrs then Today might sound correct - but i will disagree in context with it as per my understanding To(day) refers to the period post dawn of new day (post sunrise).