A nice intuitive to way to explain what a Martingale is found here (Venkat’s Computational Probability course notes). Perhaps it was known already to many. To a layman, this is the best way to introduce Martingale, first up!
A sequence is called a martingale if , for all . An intuitive way of thinking about martingales (and indeed the origins of martingales) is to imagine as your total profit after the th round of a gambling game. Then the martingale condition states that knowing your winnings in the first games should not bias your winnings in the th round. See the Wikipedia page for examples of martingales. The course notes is fabulous. I am following it on and off!