Randomness

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 or state 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.

A simple 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).

By chance
A result is said to have occurred by chance if it appears meaningful but is not because it is simply the result of a random process. For example, consider tossing a "fair coin" (50% chance of "heads") 10 times and noting the number of "heads" that are obtained. One would expect to see an equal number of "heads" and "tails", at least approximately. What if one sees all 10 tosses come up "heads"? There are two possible explanations: The second explanation might be considered unreasonable, but it is possible. (There is, in fact, a 1 in 1,024 chance of getting "heads" on all 10 tosses of a fair coin.)
 * 1) The coin is biased and comes up "heads" more often than "tails" (a "meaningful" result).
 * 2) The coin is actually fair and our results occurred merely "by chance".

See also statistical significance.

Equally likely
A set of outcomes are called equally likely if each one has the same probability of occurring as any other. When choosing one item from a group of items (the population), the item is said to be chosen randomly if all items in the population are equally likely to be the one chosen. When extending this to choosing multiple items (a sample), not only does every item in the population need to be equally likely to be chosen, but every similarly-sized set of items must be equally likely. (In particular, when choosing 2 items from 5, for example, there are 20 possible pairs of items in the population and so each of these pairs must have a 1/20 chance of being selected as our sample.) A sample chosen in such a way is called a simple random sample.

Counter-apologetics
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?"

This attempted argument from incredulity (in this case a misguided attempt to use Occam's razor) reveals a misunderstanding of the role of randomness in evolution. See Evolution is not a theory of chance.