These are the festivals of the Lord, holy convocations, which you shall proclaim at their proper time. -Leviticus 23:4
One of the things that many immigrants from traditional backgrounds learn to appreciate about living in Israel is the fact that the "second day of the holiday for the Diaspora" is not observed here. So, for example in the Diaspora, the eighth day of Sukkot, Shmini Atzeret, a full holiday, is doubled, and its second day (i.e., the 9th day of Sukkot) observed as Simchat Torah. In Israel, it is only one day, also observed as Simchat Torah, so Israel and the Diaspora celebrate Simchat Torah a day apart. Pesach is only seven days in Israel instead of eight, and there is only one seder. Most people find this a great relief (liberation!), but it causes havoc with in-law relations: instead of tonight at our place and tomorrow at yours, we have to alternate years. This is not so simple.
While of course holidays are a time of rest and joy and kedushah, I think that many Jews, no matter how devoted they are to the tradition, often find this doubling of the days burdensome, and don't miss it when they come to live in Israel.
The reason for the second day preserves the historical fact that up until about the 4th century, each new month was declared according to the observation of the new moon. Once this was ascertained, signal fires or runners transmitted the information to outlying communities - and all the way to the Babylonian and other exiles. To cover the possibility that the signal might not get through in time, an extra day was added as a margin of error: after all, everyone could see the moon and count the days - there was never a possible variation of more than a day in the declaration, so the most we could err if we didn't get the signal was one day. Within Israel this margin was not needed. Also, it was not relevant to post-Torah holidays like Purim and Chanukah, where an error would not matter so much.
But what about Rosh Hashanah and Yom Kippur? An extra day of Yom Kippur is not an option (a 48 hour fast?). And Rosh Hashanah is the only holiday that actually falls on the new moon itself - all the others come later in the month. Thus, even in Israel there was always a degree of uncertainty about the declaration of Rosh Hashanah, and so it was - and is today - observed for two days.
The Moslems still wait to sight the new moon before declaring their holidays, so while we all knew that the festival at the end of Ramadan, Id El Fitr, should coincide with Rosh Hashanah this year, we Jews knew well in advance exactly when our holiday would fall. Our Moslem neighbors couldn't tell us which days they'd be off work until the new moon was actually observed and proclaimed.
Since the fourth century our calendar has been fixed by astronomical tables, and we don't wait for the signal; the Talmud even states this explicitly. So you might think that the second day is now superfluous. And indeed, one of the first reforms of the Reform movement was to cancel it. This put the Orthodox on the defensive, as Reform had the Talmud on their side, and the main Orthodox defense was that we can't change what has been handed down to us, which felt (and still feels) a bit lame.
Meanwhile, since when Rosh Hashanah falls on Shabbat one doesn't sound the shofar or recite Avinu Malkeinu, we were rescued this year by the second day - our service on the first day was rather dry and disappointing, and we waited with anticipation for the second day. Maybe that's really why we have two days...