On August 15, 1945 Japan surrendered bringing World War 2 to an end. This day is celebrated in the UK and several other countries ( including Australia, New Zealand, the Netherlands and even South and North Korea) as V-J day, in line with V-E day which marked the end of the war in Europe.
However, the surrender document was signed on September 2. This is considered as V-J day in the US. And September 3 is V-J day in China and some other countries.
For more details see https://en.wikipedia.org/wiki/Victory_over_Japan_Day
Now we come back to the Indian sub-continent. Sometimes you may wonder how August 15 came to be India’s Independence Day.
Lord Mountbatten stated that this date came into his mind when he had to decide the date of transfer of power. This is what he said to the authors of “Freedom at Midnight”.
It also appears in numerous other references. For instance, see https://en.wikipedia.org/wiki/Independence_Day_(India)#Immediate_background
Another question is why Pakistan celebrates its Independence Day on August 14 instead of 15. In fact, for some time Pakistan did consider August 15 as its I-day and this appears on official documents (and even stamps) up to July 1948.
Naturally, the country now known as Bangladesh used to celebrate I-day on August 14 until 1971. After that it became March 26 which was when independence was declared in 1971.
There is also Victory Day on Dec 16 (when the Pakistani forces surrendered) and National Mourning Day on August 15 (marking the assassination of President Sheikh Mujibur Rehman in 1975).