A process is called random if its state at any particular time is unpredictable. The set of all possible states may be known, and the frequency of occurrence of these possible states may also be known, but the actual observed state is unknowable until you actually observe it. An example is a coin toss (specifically, one in which the coin rotates several times before landing). Such a toss results in either "heads" or "tails", and ideally these two outcomes are equally likely, but which of these will actually occur the next time you toss the coin is unknowable.
Creationists like to claim (or at least insinuate) that the modern scientific explanation of evolution is based entirely on randomness. "Which is more likely," they will ask, "That God created the great diversity of life on Earth, or that it all came about by chance?"