What is randomness? Nobody knows, or at least there’s no consensus. Everybody has some vague ideas what randomness is, but when you dig into it deeply enough you find all kinds of philosophical quandaries. If you’d like a taste of the subtleties, you could start by reading one of Gregory Chaitin’s books. Or chew on this tome.
What is a random variable? That’s easy. It’s a measurable function on a probability space. What’s a probability space? Easy too. It’s a measure space such that the measure of the entire space is 1.
Probability theory avoids defining randomness by working with abstractions like random variables. This is actually a very sensible approach and not mere legerdemain. Mathematicians can prove theorems about probability and leave the interpretation of the results to others.
As far as applications are concerned, it often doesn’t matter whether something is random in some metaphysical sense. The right question isn’t “is this system random?” but rather “is it useful to model this system as random?” Many systems that no one believes are random can still be profitably modeled as if they were random.
Probability models are just another class of mathematical models. Modeling deterministic systems using random variables should be no more shocking than, for example, modeling discrete things as continuous. For example, cars come in discrete units, and they certainly are not fluids. But sometimes it’s useful to model the flow of traffic as if it were a fluid. (And sometimes it’s not.)
Random phenomena are studied using computer simulations. And these simulations rely on random number generators, deterministic programs whose output is considered random for practical purposes. This bothers some people who would prefer a “true” source of randomness. Such concerns are usually misplaced. In most cases, replacing a random number generator with some physical source of randomness would not make a detectable difference. The output of the random number generator might even be higher quality since the measurement of the physical source could introduce a bias.