Eclectics work this way: Each golfer plays X number of rounds, and each golfer compares his scores on each hole per round. The lowest score made on each hole is recorded, producing an 18-hole score.
For example, let's say the eclectic is made up of three rounds. In the first round, the golfer scores a 6 on hole No. 1; in Round 2, he scores a 7 on hole No. 1; in Round 3, he scores a 4 on hole No. 1. The lowest of those three scores on the first hole is a 4, so that is the golfer's eclectic score.
So the golfer compares his scores on each hole in every round of the eclectic, selects the lowest per-hole score, and that is his eclectic scorecard.
An eclectic tournament can be a stand-alone event contested over consecutive days. More commonly, it is a tournament that runs concurrently, over longer time periods, with other, unrelated tournaments.
For example, the Duffersville Men's Golf Association announces a spring tournament schedule comprised of eight events. But along with those events, the DMGA offers golfers the option of also signing up for an eclectic, which runs throughout the spring schedule. In this case, the eclectic is comprised of eight rounds (or however many rounds are played over the course of the eight tournaments that make up the spring schedule) played over the course of many weeks.