What is randomness? In probability and statistics, the concept of "random" and "randomness" are frequently used.  Often the concept of a random variable is used to model events that occur due to chance.
My question regards the term "random".  What is random? Does randomness really exist?
I am curious what people that have a lot of experience in working with random events think and believe about randomness.
 A: If we assume that we are living in a deterministic (everything that happens is predetermined and given the same exact situation, the same exact things will happen), then there is no "random" at all.
In this case, "randomness" is merely used to represent what might happen given our limited knowledge.  If we had perfect knowledge of a system, nothing would be random.
A: My definition of random would be unpredictable, i.e. you can never know with 100% certainty the outcome of an event, although you might be able to put a bound of the range of possibilities. A simple example would be rolling a fair dice: you can never know exactly which number will come up with each roll, but you do know it will be one of the numbers 1 through 6.
A: You can get a really nice definition of randomness, that reflects our intuition of "unpredictability" by employing some basic concepts from information theory.
The high-level idea is to develop a concept of "compression", using some fixed "compression language".  This can be accomplished nicely in terms of the Kolmogorov complexity.  Basically, given a language used to describe things (such as English, French, etc), we can talk about compressing a source of bits.  For example, if my bit source always dumps out 01010101... where a 0 is always followed by a 1 and vice versa, then we can "compress" an arbitrarily-sized subsequence of this string:  "Write '01' 20 times" is shorter than "0101010101010101010101010101010101010101".  Let's define $L$ so that $L(0101010101010101010101010101010101010101) =$ "Write '01' 20 times"
Given this notion of compression, we can define a Bernoulli Random Variable to be an infinite sequence of bits $S$ so that, given ANY finite language $L$ (i.e. $L$ has a longest word), there exists an $N$ such that for all $n > N$, the length of $L(s_1s_2s_3...s_n)$ is greater than or equal to the length of $s_1s_2s_3...s_n$ (i.e. $s_1s_2...s_n$ is uncompressible in $L$).  Note that this is a bit weird, because the sequence is only Bernoulli "in the limit" in some sense - as we sample more and more bits, it becomes increasingly difficult to describe them without simply "listing them off".
Basically, this means that, given the tools we have at our disposal, we can't "predetermine" what the next bits in the sequence will be, unless our language is so large that we can simply include the entire sequence up to that point in our language.  For example, my language could include a one-off definition that "burbawubba := 001010100001010100100101010001010101001".  Then, I could compress "001010100001010100100101010001010101001" in my language by simply saying "burbawubba".  However, if my language is finite, I eventually have to run out of one-off encodings like this.
We can define many more distributions of random variables in terms of our Bernoulli random variable.
Now, of course, whether an actual physical source of bits that satisfy this property exist in the real world is another question entirely.  Still, we can achieve a sort of "essentially random" that is satisfactory in all the ways we would like by simply limiting the size of our language to some reasonable number.  If we limit our language to English, then we can simply go out as far as the requisite $N$, at which point the bits in the sequence essentially have no substantial relationship to anything that you can put into words.  This is usually good enough for most applications - Your sequence of coin flips can't possibly be described in terms of factors concerning the demographics or health history of that group you're sampling for medical research.  So, in that sense, it's "essentially random".
A: Here's a deflationary theory: Something is random when its behaviour is modeled formally using the machinery of probability theory, an axiomatized bit of pure mathematics.  So in a sense the answer to the first question is rather trivial.
In approaching the rather less well-posed question 'does randomness really exist?' it's helpful to ask yourself whether vectors 'really' exist.  And when you have a view about that, asking yourself a) whether it's surprising or not that polynomials are vectors, b) whether and how we could be wrong about that, and finally c) whether, e.g. forces in physics are the things that vectors 'are' in the sense of the question.  Probably none of these questions will help much understanding what's going on in the forum, but they will bring out the relevant issues.  You might start here and then follow up the other Stanford Encyclopaedia entries on philosophy of probability and statistics.  
There is a lot of discussion there, thankfully not much found around here, about the existence and relevance of 'actual' physical randomness, usually of the quantum variety some of which is (usefully) gestured toward by @dmckee in the comments above.  There's also the idea that randomness as some sort of uncertainty.  Within the minimal framework of Cox it can be reasonable to think of (suitably tidied up) uncertainties as being isomorphic with probabilities, so such uncertainties are, by virtue of that connection, treatable as if they are random.  Clearly the theory of repeated sampling also makes use of probability theory, by virtue of which its quantities are random.  One or other of these frameworks will cover all the relevant aspects of randomness that I've ever seen in these forums.  
There are legitimate disagreements about what should and should not be modeled as random, which you can find under the banners Bayesian and Frequentist, but these positions only suggest but do not full determine the meaning of the randomness involved, just the scope.
A: I tend to prefer a probabilistic interpretation of randomness.  An event is random if gaining any additional information does not help you predict its outcome.  That is, the event is unconditionally random. Notationally:
$p(A|B) = p(A) \forall B$
To put it in concrete terms; if you believe that a die roll (A) is truely random, then knowing the exact physical state of the die as it is thrown (B) confers no additional predictive power on the outcome of the toss.
A: One of my favorite interpretations is the sampling vs design based uncertainty described in this paper.

*

*Sampling uncertainty comes from the fact that a researcher collects a survey at random from the population. Randomness of the estimators such as the means comes from the fact that the researcher could have collected many different subsets of data. "Ex-ante" it is unclear which individuals will be chosen as part of the survey.

*Design based uncertainty arises in randomized controlled trials because the researchers randomly allocate individuals to treatment and others to the control group. For each individual there is a potential outcome if she receives the treatment and if she doesn't. The difference between both groups is random because different sets of people could have been assigned to either group.

In both cases "randomness" is used to model hypothetical scenarios where different people would be chosen to be sampled/treated, even if the underlying outcomes of the "superpopulation" are fixed. This thought process is extremely important, because analysts want to draw conclusions that are not particularly sensitive to the sample chosen.
