In classical physics, events are random only due to insufficient information. Let’s take the example of a coin toss. With an ordinary coin, there’s a 50% chance of heads and a 50% chance of tails. We say that this outcome is “random.”
But if we had enough information about the initial position of the coin, the forces that our fingers exert on it, the air resistance, the distance it must fall before landing, and so on–with sufficient information, we could predict the outcome. The outcome wouldn’t be random at all. The problem is really that the causative factors are too hard to measure. They are hidden to us. Physicists call these hidden causes “hidden variables.” In summary, in classical physics, events which appear to be random are actually just a comment on our ignorance. Randomness due to ignorance is called “classical randomness.”
In quantum physics, many physicists believe that some events at the quantum level really ARE random. For example, the moment that a particular atom of uranium will decay due to natural causes appears to be random. It’s as if nature installs a random number generator into each atom of uranium, and the nucleus splits when its number comes up. This is called “quantum randomness” or “true randomness.” For more information see the entries for acausal and quantum randomness.