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.