What does opening the door for Elijah symbolize?
In my family, and many others, I am sure, welcoming Elijah into the house on Passover also means offering him a glass of wine, which we always referred to as "Elijah's Cup." We fought over the honor of opening the door for him. Elijah's cup is always a fancy goblet, set in the middle of the table, filled to the brim with wine for him. Historically, there have been rabbinical disagreements over whether Jewish people are supposed to drink four or five glasses of wine during the Passover meal, the Seder, and this fifth glass has turned into Elijah's cup, a fifth cup, a way of hedging on what the requirement really is supposed to be. When I was a very young child, maybe four or five, on one Passover at an aunt and uncle's house, I swear I saw the level of wine go down in Elijah's cup after the door was opened. My family has always been quite skeptical about this. Of course, imagine the effect of even a few sips of wine on a small child!
I am assuming your question is referring to the Jewish Passover celebration in which participants symbolically open the door, welcoming the prophet Elijah to enter the home. An extra place at the table is also set in anticipation for the guest. These symbolic actions concerning the anticipation of the prophet Elijah are done every year at the Jewish Passover.
This expectant and hopeful anticipation of the prophet Elijah hearkens back to a prophecy in the Jewish book of Malachi, found in the Old Testament of the Christian Bible. In Malachi, the God of the Hebrews says, "Behold, I send My messenger, and he will prepare the way before Me" (Malachi 3:1, NKJV). The Jews understood this to be Elijah, who would appear to herald the coming of the Jewish Messiah. Jews, anticipating the coming of Messiah for thousands of years, still today anticipate the prophet Elijah, who announces the coming of Messiah, by setting a place at the table and opening the door for his entrance.