There is a widespread custom to eat a hardboiled egg at the Pesah Seder, in order to commemorate the Korban Hagiga (the "festival sacrifice"). During the times of the Bet Ha'mikdash, the Hagiga offering was brought together with the Korban Pesah (paschal offering) on Erev Pesah, and one would partake of its meat prior to eating the meat of the Korban Pesah. We commemorate this practice by eating an egg, the traditional food of mourning, in order to express our sorrow over the absence of the Mikdash, on account of which we are unable to fulfill the Misvot of the holiday as we should.
An interesting question arises regarding this Halacha in a case where Erev Pesah occurs on Shabbat, such that the Seder is held on Mosa'e Shabbat. During the times of the Mikdash, when Erev Pesah fell on Shabbat the Korban Hagiga was not offered on that day. Although the offering of the Korban Pesah overrides the Shabbat restrictions, this is not true of the Korban Hagigah, and thus when Erev Pesah fell on Shabbat the Korban Hagigah was not brought together with the Korban Pesah. Perhaps, then, in such a case we should not partake of an egg at the Seder in commemoration of the Hagiga sacrifice.
Tosefot address this question in Masechet Pesahim, and they conclude that one must, in fact, eat an egg at the Seder even on Mosa'e Shabbat. They cite Rabbenu Peretz (France, 13th century) as explaining that if we would not eat an egg in this case, we would give the impression that in other years we actually partake of the Hagiga sacrifice, rather than a symbolic commemoration. This impression was especially likely in communities that would eat meat rather than an egg to commemorate the Korban Hagiga. In order to avoid giving this impression, we eat the egg at the Seder even in situations where the Korban Hagiga would not be offered, thus making it clear that the egg serves merely as a symbolic commemoration.
The Rosh (Rabbi Asher Ben Yehiel, Germany-Spain, 1250-1327) cites a different theory in the name of Rabbi Yishak of Courbeil (the "Samak," France, 13th century). He suggested that one should eat an egg at the Seder even on Mosa'e Shabbat because when Erev Pesah occurred on Shabbat the Hagiga was offered on Friday. Since the Hagiga was brought even in years when Erev Pesah fell on Shabbat albeit on the previous day it is appropriate to include a commemoration of this sacrifice at the Seder. The Rosh, however, rejects this theory, noting that when the Hagiga was brought on Friday, its meat would no longer be permissible for consumption on Mosa'e Shabbat. Seemingly, then, there should be no reason to commemorate the Hagiga at the Seder on Mosa'e Shabbat.
In any event, in light of the comments of Tosefot cited above, Halacha requires partaking of a hardboiled egg at the Seder even on Mosa'e Shabbat. This is indeed the ruling of the Ben Ish Hai (Rabbi Yosef Haim of Baghdad, 1833-1909), the Kaf Hahayim, and the Mishna Berura (commentary to the Shulhan Aruch by Rabbi Yisrael Meir Kagan, 1839-1933).
Summary: One should eat a hardboiled egg at the Seder to commemorate the Hagiga sacrifice, even when the Seder is held on Mosa'e Shabbat.