The prophet Elijah is the person whose appearance is meant to bring in the coming of Moshiach, the Messiah. In the Seder service, Elijah's role has expanded greatly.
At one point after the meal, the front doors to the house are opened and any strangers who want to may come in and partake in the Seder and the meal. This custom comes from times when Jews were falsely accused of drinking Christian's blood instead of wine, and any passers-by were welcomed in to see the truth (incidentally, Pesach wine even used white grapes for a while to further dispel the myth). Nowadays, as well as serving it's original purpose, folklore has seized upon the custom to say that we open the door for Elijah, who comes in and drinks from his goblet of wine. Parents around the table knock the table gently to fool children into believing that Elijah is taking a drink.
There is also sometimes a place setting left aside for Elijah, or for any passer-by, to come in and eat.