Why do we use a roasted (maybe?) egg on the Seder Plate to represent the Festival offering?
The Shulchan Aruch Harav says that one eats eggs to remind him to mourn the Beis Hamikdash, since had the Beis Hamikdash stood, we would have been eating the korban Pesach.
The Taz (OC 473:4) suggests the reason is so that we can eat it, because it is forbidden to eat roasted meat on the Seder night.
The Mishna Berura (:23) quotes two other reasons. First, that we use an egg, which is commonly served at a meal to mourners, to represent our mourning the loss of the Beit HaMikdash and the Korbanot. Second, the word for egg in Aramaic is ביעא which is symbolic of the phrase בעי רחמנא למפרק יתנא (a plea for redemption).
My personal favorite explanation is due to the Chatam Sofer: The egg represents the Jewish people. Most foods become softer as they are cooked. But the egg becomes harder. So it is with the Jewish people: The more they are oppressed, the stronger they become.
Note that we don't have an ancient explanation for the egg. The first known reference to it is by Rabbi Yitzhak ben Abba Mari in late 12th-century Marseilles, France. [Sefer Ha'ittur 2:133c] Speculation on the reason has abounded since that time.