For two centuries after Pascal and Fermat, probability theory was the study of events — coin lands heads, card is a queen, two dice sum to seven. The arithmetic worked, but the theory could not easily marry the calculus that the rest of mathematics was developing in parallel. The breakthrough was a notational move so simple it can be missed: stop talking about the event itself and start talking about a number that depends on the outcome. Define the number of heads in ten flips, or the sum of the dice, or the time until the next bus arrives, as objects in their own right. Once you do, the apparatus of calculus — derivatives, integrals, limits — pours into probability through the open door, and the field becomes the unified subject we now teach.
Formally, a random variable X is a function from the sample space Ω to the real numbers, with the technical requirement that the preimage of any reasonable subset of ℝ is a measurable set in Ω. The technical requirement is a measure-theoretic nicety that ensures probabilities can be assigned consistently; the conceptual content is just a number you can read off once the experiment is run. Random variables come in two main flavors. Discrete random variables take values in a countable set: the number of heads in N flips, the count of customers arriving in an hour, the suit of a drawn card encoded as 1–4. Continuous random variables take values in a continuum: the height of a randomly chosen person, the wait time until the next earthquake, the fluctuating price of a stock. Either kind is described by its cumulative distribution function F(x) = P(X ≤ x), which gives the probability that the variable falls below any given value. From the CDF you derive the probability mass function (discrete) or probability density function (continuous), and these capture the full statistical content. The deep payoff is that all of analysis applies: you can take expectations (which are integrals), compute variances (more integrals), prove convergence theorems, and bring the entire machinery of calculus to bear on uncertainty. Without random variables, the Central Limit Theorem cannot even be stated, let alone proved.
Every applied probability model in modern science and engineering is built around random variables. Particle physics simulations model decay times, scattering angles, and detector noise as RVs. Machine-learning loss functions are expectations of random variables (the loss as a function of a random training example). Reliability engineering treats time-to-failure as an RV with a chosen distribution. Financial models treat returns, volatilities, and shock arrivals as RVs. The conceptual move — from event to number-that-depends-on-outcome — is the bridge that lets probability participate in the rest of quantitative science.