When we use the word "elect" we are often referring to a person who has recently been elected to an office, whether it be locally or at the state or federal level. They are referred to as the elect in the time between when they win an election and when they actually take office. Thus, right now, all of the new senators and legislators who are about take office in January 2011 after the November 2010 elections are called Senator Elect Smith (or whatever their last names are).
There are many forms of the word elect. The way you refer to it in your question suggests this form of the word above that I have suggested which is a noun. To elect someone would be to vote for them with is an action and thus a verb. The electorate are those who vote. The election is the act (another noun) or event held which allows voters to cast their ballots.
It would be helpful if you would tell us what context you are asking this question in. However, the most likely answer for this is that the term "the elect" refers to those people who have been picked by God to be saved (brought to Heaven) after they die. The idea of the elect was a Puritan or Calvinist idea.
The Calvinists (Puritans were Calvinist) believed that God had "predestined" people to go to Hell or Heaven. People could not affect their destiny -- nothing they did could change where they were going after they died.
In this way of thinking, the elect were those who had been chosen to go to Heaven.