1. On Our Clocks, the Date Changes at Midnight
In everyday life, a new date begins at midnight, and the day lasts until the next midnight. On a 24-hour clock, 00:00 is the start of a day and 24:00 is the end of the same day. So "Friday 24:00" and "Saturday 00:00" name the same instant, written as the end of one day or the start of the next [1].
The trouble is the words. Midnight is twelve hours before noon and also twelve hours after noon, and noon is neither "before noon" nor "after noon." For that reason the U.S. National Institute of Standards and Technology (NIST) says that "12 a.m." and "12 p.m." are ambiguous and should not be used. If you want to be clear, write 11:59 p.m. or 12:01 a.m. with the date, or use the 24-hour clock [1].
That makes a good test for a plan like "meet at 12 a.m. on Saturday." Does it mean the night between Friday and Saturday, or the night between Saturday and Sunday? Different people read it differently, which is why writing the date and using 11:59 p.m. or 12:01 a.m. is safer.
2. Astronomers Once Changed the Date at Noon, Not Midnight
Midnight was not always the line. For astronomers, the middle of the night is prime working time, because stars are visible and a night of observing is one long job. If the date flipped at midnight, a single night's notes would be split across two dates. So for a long time astronomers counted days from noon to noon, and each night stayed under one date [3].
Even the famous clock at the Royal Observatory in Greenwich once showed this kind of time. Royal Museums Greenwich says the clock "originally indicated astronomical time, in which the counting of the 24 hours of each day starts at noon," and that it was changed in the 20th century to show Greenwich Mean Time, which counts from one midnight to the next [2].
According to the encyclopedia article on Greenwich Mean Time, the midnight convention was adopted for astronomical purposes on and after 1 January 1925. That caused a jump of half a day: the instant labeled "December 31.5" in a 1924 almanac became "January 1.0" in a 1925 almanac [3]. This exact date rests on that one encyclopedia article, so treat it as reported rather than checked against the almanacs themselves.
So the answer to "when does the day start?" is partly a matter of who is asking. A farmer, a sailor, a lawyer and an astronomer have each had good reasons to put the line somewhere different.
3. The International Date Line Means the Date Changes at Different Moments Around the World
Even if everybody agrees on midnight, midnight arrives at different times in different places, because Earth turns and it is night in one place while it is day in another. So people have drawn a line on the map where, by convention, one calendar day is treated as ending and the next as beginning. That line is the International Date Line, a line from the North Pole to the South Pole that runs roughly along 180° longitude in the Pacific Ocean and bends around some islands and countries. Cross it traveling east and the date goes back by one day; cross it traveling west and the date goes forward by one day [4].
The line is a map convention. It is not defined by international law, and countries whose waters it passes through have moved it when it suited them [4].
The clock times on the two sides can be very far apart. Kiribati's Line Islands use UTC+14, and uninhabited Baker and Howland Islands use UTC−12. That is a 26-hour gap between their clocks [4].
Here is a thought that follows from those numbers. If those two zones are the extremes, a new date begins in the Line Islands first, and it begins at Baker and Howland up to 26 hours later, where it then lasts 24 hours. So the same calendar date could be "alive" somewhere on Earth for about 26 + 24 = 50 hours. That 50 is my own calculation from the 26-hour gap, not a number printed in a source.
4. Countries Have Moved the Date Line by Skipping Days
Because the line follows human choices, a country can decide to switch sides. When it does, it must skip or repeat a date on its calendar.
Kiribati is a nation of islands spread across the line. Its territory lay on both sides of the line, so the two halves of the country were on different dates, which made working together awkward. In 1994 its eastern islands skipped Saturday, 31 December, so that Friday, 30 December was followed by Sunday, 1 January 1995. Effectively, the line was bent east around the whole country [4].
Samoa made the opposite kind of move in 2011. Samoa's calendar was a day behind those of nearby Australia and New Zealand, and the change was meant to remove that one-day offset. So Samoa removed Friday, 30 December 2011 from its calendar. Thursday, 29 December was followed directly by Saturday, 31 December [4].
If you lived in Samoa that year, there was no 30 December at all. The day that never happened is a good reminder that a date is something people agree on, not something nature hands us.
5. Try It: Follow One Date Across the World
With an adult, pick a day and look at a world clock website or phone clock for Auckland, London and Los Angeles. At one moment, which cities say it is already tomorrow? You will see the new date arriving in one place after another, like a wave moving west. The date line is where the wave starts again.
You can also look at a TV schedule or timetable that uses a 24-hour clock and find 00:00. Is it written at the end of one day or the start of the next? Then try writing your own plan using "11:59 p.m. Friday" instead of "12 a.m." and see whether a friend reads it the same way.