How do Jewish calendars around the world know when the Jewish holidays fall out each year? Is there a certain beit din that has kept track of the 19 year cycle?

According to the halacha, a Beit Din of ordined Rabbis ("semuchim" or "mumchim") must declare the new moon which is the beginning of the calendar month. This ordination ("semicha") is only possible in Eretz Yisroel. During the Talmudic period, with the transfer of the centers of learning to Bavel, it became impossible to maintain the smicha in Eretz Yisroel. At that time the sanhedrin set up a calendar which sets out the 19 year cycle and the rules of maintaining the months and holidays from that time forward. That sanhedrin actually invested the calendar with the authority of "semuchim. It is not necessary for a subsequent beit din to approve the calendar, as the rules are already set out. The calendar is therefor standard for all Jewish communities. (See Rambam "Mishneh Torah - Hilchot Kiddush haChodesh" ch. 5)