A breakdown of the Jewish holidays and festivals, and why some are celebrated for an extra day outside of Israel.
Jewish life is measured by the Hebrew calendar, which is based on a solar year comprising 12 lunar months. To correct the discrepancy between the lengths of the solar and lunar years, an extra month is added to the calendar every two to three years (according to a complex calculation that works out to seven of every 19 years being a leap year).
Although most Jews live in countries that use the Gregorian calendar, Israel included, the timing of Jewish festivals and holidays, from Rosh Hashanah to Tisha B'Av, is determined by the Hebrew calendar.
The Hebrew day lasts from sunset to sunset, which naturally means that holidays begin - and end - in the evening, when the sun sets. Also, each Jewish month begins with the appearance of the new moon. In ancient times, this depended on actually sighting the sliver of the new moon in Jerusalem.
Because people outside Israel couldn't always know when exactly that happened, to be safe, they celebrated certain holidays for two days. This custom is still observed today by Jews living in the Diaspora.