Elijah, a biblical prophet, has special significance in several Jewish religious rituals, including both circumcisions and Passover seders; Elijah's invitation to attend seders is related to this special significance. According to Chabad.org, Elijah is invited to attend Passover seders because he has the unique ability to confirm that the men in attendance are circumcised, meaning they are fit to partake of the Passover ritual.
Elijah's presence in the ritual can be felt in a few different ways, from a designated cup of wine to a full empty place setting reserved for the invisible presence of the prophet. In most situations, seder attendees will open the front door of the residence or building in which the seder is being held in order to invite Elijah in to join the celebration. This may happen during the seder or after it has finished.
As with most aspects of Passover celebrations, Elijah's significance in the meal may depend on the level of strictness the people hosting or attending the seder tend to observe when approaching religious rituals. More informal seders may include guests who aren't Jewish, and Elijah's significance as an observer of religious circumcision may be less of a factor in the Passover celebration.