Short answers, pulled from the story.
A random variable is a measurable function from a probability measure space, called the sample space, to a measurable space. Despite its name, the term in its mathematical definition refers neither to randomness nor variability; it is a precise function that translates outcomes into values, usually real numbers.
According to George Mackey, Pafnuty Chebyshev was the first person to think systematically in terms of random variables.
A discrete random variable takes values in a finite or countably infinite set and is described by a probability mass function. A continuous random variable has a continuous cumulative distribution function with no gaps, and each exact value has probability zero; probabilities are assigned to intervals using a probability density function.
A mixed random variable is one whose cumulative distribution function is neither purely discrete nor everywhere continuous. It can be realized as a mixture of a discrete part and a continuous part, and every probability distribution on the real line is more generally a mixture of a discrete part, a singular part, and an absolutely continuous part.
The expected value, also called the first moment, is the mathematical formalization of a random variable's average value. It can be understood intuitively as an average obtained from an infinite population whose members are particular evaluations of the variable.
Two random variables are equal in distribution if they share the same cumulative distribution function, even across different probability spaces. They are equal almost surely if the probability that they differ is exactly zero. Variables equal in distribution but not almost surely can have different covariances with a third variable defined on the same probability space.