I know there are still other calendars, like the Jewish calendar and Islamic calendar, but for the most part everyone uses the Gregorian calendar that includes February 29th as the leap day.
When and how did this international acceptance come about?
The ancient Romans originally had ten months, the forerunners of March - December. (this is the reason the later months have names which correspond to latin numbers two less than you would expect). Originally the season of Winter was outside the calendar, its days were undated, but eventually January and February were invented to allow all days to have a calendar date.
There was now a problem, however. Remember that a Month lasts around the time it takes for a Moon to turn full then turn dark again. However, it does not do this an exact number of times during the year. So some years, on an ad-hoc basis, they would end February a few days early and have a whole new month called Intercalaris in their place. March would then follow Intercalaris.
A problem was that there was no set mathematical system for deciding this, instead a religious official would make the decision. Now, imagine this was happening today and the official was a Republican while the current leader was a Democrat. Since they'd want to keep the leader's term as short as possible, the temptation is to decide not to add the Intercalaris month. Now imagine both the official and the President are Democrats. The temptation would be to add the Intercalaris to keep him in power longer.
When Julius Caesar took power, he shook quite a few things up. One of them was to end the idea that a Month correspond exactly to the lunar cycle. Instead of adding a whole month after February (which, remember, is the twelfth month if you start counting from March), he simply lengthened a festival towards the end of February by one day, every four years. Technically, the day he added was the 24th day of February, not the 29th, but in modern times we only think about the numbers rather than the ancient Roman festivals.
Technically they made an error and added the leap day every three years instead of every four, they realised their mistake after about a generation and corrected it.
This calendar was the Calendar of the Roman Empire, so as the Romans spread across Europe it was introduced to places that previously had no calendar. When the Roman Empire fell, the calendar remained among its former subjects.