elect puritans are puritans who have been predetermined by god to be saved. Puritans believed that their destination was predtermined by god and that the elect or saints were to be saved
The Puritans followed the theological teachings of John Calvin, who reasoned that since God, by definition, knows everything, he must know who is ultimately going to go to heaven and who is going to hell, before they are even born.
This concept is called "predestination". So the "elect" are those of us who are heading for heaven, and the "unregenerate" are those who will go to hell.