Formally a measurable function from the sample space to the reals. Practically: the numeric thing you are uncertain about.
Discrete random variables take countably many values and are described by a probability mass function. Continuous ones are described by a density, where P(X = x) = 0 for every single point and only intervals carry probability.
The trap. A density can exceed 1 - it is a density, not a probability. A uniform distribution on [0, 0.5] has density 2 everywhere on that interval, which surprises people who read f(x) as a probability.