A process is called random if its state at any particular time is fundamentally unpredictable (that is, not just unpredictable as a practical matter). The set of all possible states may be known (this is typically called the sample space), and the frequency of occurrence of these possible states may also be known (this can often be determined using ideas of probability), but the actual observed state is unknowable until you actually observe it.
An example is a "well tossed" coin (i.e., 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.
Another, perhaps more interesting, example might be the adult height of the offspring of two people. Knowing the heights of the parents, and perhaps even the grandparents, along with the general socio-economic status of the family (since poor health can affect growth), might help to estimate the final, adult height of the child, but such an estimate is mostly just a guess; the actual height is fundamentally unknowable until the child actually grows up.
Both examples above can be modeled mathematically by specific probability distributions, which characterize both the set of possible outcomes and their corresponding probabilities. Specifically, the coin-tossing example can be modeled by the so-called Bernoulli distribution and the height example by the Normal distribution (specifically, in the context of a multiple regression model).
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?"