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According to biblical law, every Yom Tov is just one day. In fact, this has always been the practice in the Land of Israel. However, the Sages ordained that an additional day be celebrated in the diaspora, called Yom Tov Sheni (the second day of Yom Tov).
Observance of the second day began back when the Sanhedrin sanctified each month based on eyewitness testimony. Once this was done, messengers would depart to spread the news, so people everywhere would know when to celebrate the upcoming festival. However, the messengers could not reach diaspora communities before the festival. Therefore, the standing instructions for the diaspora were to keep each Yom Tov for two days on account of the uncertainty about the date, so everything done on the first day would be repeated on the second day. For example, in Israel there was (and is) a Seder on the first night of Passover, while in the diaspora there was (and is) also a Seder on the second night.
Even after the calendar was set (section 1 above), the rule was not changed. There are technical reasons for this. There is also a spiritual reason, as halakhic reality reflects spiritual reality. In the Land of Israel, holiness is more accessible, and therefore the holiness of the festivals can be absorbed in the one day the Torah assigns. In contrast, those in the diaspora are further away from the source of holiness, so they need two days to absorb the holiness of the festivals. This can be compared to a flashlight. When a flashlight is pointed at a nearby location, the light is strong and focused on a small area. Alternatively, when the flashlight is pointed at a distant location, the light is weak and diffused over a large area. So too, in the Land of Israel the illumination of the festivals is focused and concentrated into one day, while in the diaspora the light is weaker and diffused over two days.
The Sages declared that if one leaves the Land of Israel for the diaspora and plans to remain abroad for a number of years, one is considered a diaspora resident and is obligated to keep two days of Yom Tov. However, if one is only visiting and intends to return to Israel, one retains his basic status as a resident of Israel. Therefore, residents of Israel who are somewhere in the diaspora without a Jewish community, do not have to observe a second day of Yom Tov, and they may do melakha. Yet, if they are in or near a Jewish community, they must make sure not to be seen deviating from the local practice on the second day, which would ruin the Yom Tov atmosphere.
Someone who is visiting Israel from the diaspora is still a diaspora resident and must observe the second day of Yom Tov. Visitors who have a significant connection to Israel, and there is some chance they will make aliya at some point, are considered Israeli when in Israel and observe only one day of Yom Tov. Therefore, someone who comes to Israel to study for a year is considered a resident while in Israel, as is someone whose visits to Israel cumulatively add up to a year. Those who plan to make aliya in the future, who have bought an apartment in Israel for visits, or whose parents or children live in Israel, are considered to have a significant connection to the Land, and should observe one day of Yom Tov while in Israel for a festival. Nevertheless, until they actually move to Israel, they must continue to observe two days of Yom Tov while in the diaspora.
