What is the purpose of Election Day in Hawthorne's The Scarlet Letter?

The purpose of the Election Day celebration is to inaugurate a new man as governor of the Massachusetts Bay Colony.  Hester explains to Pearl that all the people have gathered in order to see the procession consisting of the governor and magistrates and ministers, with soldiers marching before them and music playing.  She says that "a new man is beginning to rule over them" and, as is typically the custom since the first nation was founded, everyone has come together to make merry as though in anticipation of a "golden year" about to begin under this new man's leadership.  It is, frankly, a rather optimistic time by Puritan standards. 

However, in terms of the novel, Election Day is a good reason to have everyone out and about in the town, just as they were when Hester was first shamed upon the scaffold some seven years prior.  Children are out of school, people have left their homes and businesses, and even some of the Indians have come to town to see the goings-on.  This equivalent scene bookends the story: it began with Hester's public humiliation and ends with Dimmesdale's.  Just as she was called on to speak her guilt before the entire community, so will he speak his guilt before the entire community.  The Election Day crowd neatly parallels the crowd who gathered at the spectacle of Hester's humiliation, and it allows Hawthorne to give Dimmesdale a scene as public and significant as hers had been.

What was "Election Day" in The Scarlet Letter?

The answer to this question can be found in Chapter 21 of this novel, which describes Election Day, which is the annual installation of magistrates. In this ceremony, the narrator tells us, something of the majesty and importance of ceremonies in old England is captured and there is much pomp and circumstance:

The fathers and founders of the commonwealth--the statesman, the priest, and the soldier--deemed it a duty then to assume the outward state and majesty, which, in accordance with anitque style, was looked upon as the proper garb of public or social eminence. All came forth, to move in procession befoe the people's eye...

In addition we are told that the people are allowed to relax their usual Puritan habits and customs, and to enjoy themselves more than would be normally publicly sanctioned. The Election Day is therefore an important festival in the Puritan calendar, and of course is particularly important in this story as Dimmesdale plans to preach his last sermon here before leaving with Hester to start a new life.