An event is random if it has multiple possible outcomes that can't be accurately predicted, even though it may be possible to calculate the probabilities of individual outcomes. For example, rolling a die is random because it isn't possible to predict which side will land face up, but the probability of any given side landing face up is known (being the reciprocal of the number of faces).
Randomness is a tool in role-playing game design that adds uncertainty to a situation, thereby increasing tension and, hopefully, engagement among the players.
Randomness is used in many different parts of role-playing, including: