Of all the events and scriptures associated with Elijah, the one that left the deepest impression upon the Jews is Malachi 4:5-6 (or 3:23-24 if you are using the masoretic text).
Behold, I will send you Elijah the prophet before the coming of the great and dreadful day of the LORD: And he shall turn the heart of the fathers to the children, and the heart of the children to their fathers, lest I come and smite the earth with a curse.
Elijah was identified with the messenger of the covenant in 3:1, not only on the strengths of a logical comparison of the last two verses with the first one, but also because of Elijah's zealous struggle to uphold the covenant (1 Kings 19:14). This was the opinion of the early sages.
Based on Malachi 3, they saw the role of the messenger of the covenant as one of resolving religious disputes among the faithful, upholding the oppressed, punishing the oppressors, restoring things to their proper place and order, and puting down contention.
Rabbi Yehoshua said:
I have received this from Raban Yochanan ben Zakay (Ribaz), who heard it from his teacher, who heard it from his teacher, as true as the instruction from Moses at Sinai, that Elijah comes not to defile or purify, to draw near or push away, but to push away those who are near because of strength of arms (violence, threats, coercion, strongarming, corruption, that sort of thing), and to draw near those pushed away because of strength of arms....
Rabbi Shimon says: to resolve disputes...
And the sages say: not to push away and not to draw near, but to bring peace to the world, as it says: "I will send you Elijah the prophet", and concludes with "And he shall turn the heart of the fathers to the children, and the heart of the children to their fathers ".
Mishnah, m. Eduyot 8,7.
We now reach, in a round-about way, the subject of this post. Elijah's seat.
In chapter 29 of Pirkei Derabi Eliezer, a psuedopigraphic work dated by most scholars to sometime around the Muslim conquests, several legends are related regarding circumcision. Circumcision was the sign of the covenant between the Lord and his people, and it was practiced by all the house of Israel until they split int two kingdoms. The kingdom of Ephraim (Israel) stopped its citizens from circumcising, which caused the covenant to be broken. Elijah arose in a fit of jealous (zealous, the two words in Hebrew are identical) rage, and swore the heavens to let no dew or rain fall upon the land.
As a result jezebel tries to kill Elijah. He prayed unto the Lord, who asked him if he were better than his fathers, listing many, from Jacob to David, who were forced to flee for their lives. Elijah gets the not-so-subtle hint, and takes off into the wilderness. Here the Lord again speak to him, Elijah says that he has been zealous for the sake of the covenant, and the Lord replies that he has ever been zealous.
By your life (an oath), Israel shall not circumsize a soul unless you behold it with your own eyes.
As a result, the sages made a seat of honour for the angel of the covenant, as it is said (Mal 3:1): and the messenger of the covenant whom ye delight in cometh. The God of Israel shall hasten and bring in our lifetime a messiah to comfort us.
In Sephardic and Eastern synagogues stands a special chair, Elijah's chair.
Whenever a boy is circumcised, before being given to the godfather, he is placed in that chair, to be held by Elijah, who is present, but unseen.
I myself was circumcised (whole other story, for a different post, at a later date), it was in a Morrocan synagogue, and I too was placed in that seat.
Personally, I rather like the symbolism. It is through the agency of Elijah that we, as latter-day saints, are able to enter into some of the most precious and most sacred of covenants with the Lord, covenants that truly turn children's hearts to their fathers.