Past occurrences of an independent random event have no impact of future occurrences. For instance, the probability of getting heads in a coin toss is 1/2 regardless of the previous tosses. If we encounter a dozen heads in a row, we may feel that the next toss will be tails to balance things out. This is a mistaken belief--the probability of tails remains 1/2, the same as in previous tosses.
A game of roulette at Monte Carlo Casino on August 18, 1913 is perhaps the most iconic example of the fallacy. The ball fell in the black 26 times in a row causing gamblers to lose millions in the mistaken belief that the probability of red was now higher.