Translation:The Passover Seder is the first night of the Passover holiday.
Because the first and last days of the holiday are celebrated similarly to Shabbat (no work, etc.) and while in Israel there is only one such day at the beginning followed by another one at the end, outside of Israel there are two such days at the beginning and two such days at the end. The holiday of Sukkot follows the same pattern
In fact every holiday in the diaspora is "doubled" and is referred to as יום טוב שני של גלויות. It is a fence made by our sages as the new moon was declared in Israel and it took time to spread the word to other regions. Hence they created a one day "buffer zone" (ספיקא דיומא). The only exception to this will be the fast days and Shavuot which is only counted as 50 days from Pesach, without a fixed date.
The comments are correct, but the real reason it's done this way is they are doing what used to be done to make sure they celebrated the holiday on the correct day (from communities not close to the Temple, so they couldn't make sure they had the right day). So they added an extra day to make sure.
Xerostomus, I can see our very brief and nested comments are confusing - let me try and reconstruct the thoughts behind this conversation. AnCatDubh said וְהַמִּימוּנָה הִיא הַיּוֹם הָאַחֲרוֹן - Mimounah is the last day. Now as I understand it, although it doesn't exist in my family's (Ashkenazi) tradition, it's a celebration of the return to חמץ on the day AFTER Pesach ends. Telstar made that point, and AnCat replied "potato potahto" which I took to mean "there's no essential difference, it's just two equivalent ways of saying the same thing." So I responded that for observant Jews, who do keep the Passover rules for 8 days, there's a big difference between the last day of doing that (היום האחרון ) and the day after it's all over, when we start eating regular (leavened) bread again -- whether or not we call it Mimounah. Does that answer your question?
Is there a more literal translation of ליל הסדר? Does it just mean "night of the seder"? I'm guessing ליל is לילה when it's the first word in construct state, is that correct?
So if I'm correct, is the Passover seder referred to in Israel as "night of the Seder," or is this really saying something more like "the night of the Seder is the first night of the Pesach holiday"?
The Passover Seder was originally called סדר פסח, and it is still used, but also commonly referred to as הסדר ("the Seder"). The night in which the Passover Seder occurs is called ליל הסדר - literally "the night of the Seder", as you wrote. סדר פסח and ליל הסדר are not identical: סדר פסח or הסדר is the ceremony, ליל הסדר is the night in which the ceremony is performed.
In Hebrew ליל הסדר is the official name of the event but it is mostly called just הסדר. It is somehow like the Israeli currency which is officially called New Israeli Shekel but everybody calls it Shekel or more rarely nowadays Shach (the Hebrew initials of Shekel Chadash).