The word 'seder' is Hebrew for 'order'. The Jewish Seder is the ceremonial meal in Passover, during which, according to a set of ancient customs and text, we recount the Exodus from Egypt.
Reform Jews have only 1 seder instead of 2.
Generally it occurs only during Passover, a Jewish holiday. There is also a seder for the holiday of Tu Bishvat, but this seder is rarely observed.
Yes
The Jews eat the Passover Seder meal on the night of Passover (Pesach). It makes little difference whether Passover begins on Shabbat or on a weekday. Note that there are festive meals on every Shabbat, but they are not called "seder" and the foods are different.
The Haggadah contains the service for the seder.
At the Seder meal we retell the story of the Exodus from Egypt.
Outside of Israel, Orthodox and Conservative Jews have 2 seders, but Reform Jews only have 1. Inside Israel, everyone has one Seder.
It symbolizez the bitterness of life in ancient Egypt for the Jews.
The Seder Plate is a special plate for holding the symbolic foods used during the seder. Pieces of the symbolic foods are distributed to everyone during the course of the seder. Traditionally, the person leading the seder is the only one with a Seder Plate. However if there are many guests, additional Seder Plates can also be set on the table.
In Israel, the Seder is on the first night of Passover. Outside Israel, Orthodox and some Conservative Jews celebrate Seders on the first two nights of Passover. Most other Jews have a Seder on only the first night, though some may also have Seders on the first two nights of Passover.
They can, but it is a big honor to be invited. Especially if you are not Jewish.
Moses and Eliyahu (Elijah) are each mentioned once in the Passover Haggadah. Symbolically, Eliyahu comes to the Seder and we prepare a cup in his honor. He symbolizes the future redemption (Malachi ch.3). See:More about the Seder