the law (distribution) of a random variable

When you build a random variable X, you start with an underlying experiment that already has a way of assigning probabilities to its outcomes. The variable X then pushes that probability over onto the number line: the chance that X lands in a set A of numbers is just the chance the original experiment produced an outcome that X sends into A. The result, this assignment of probability directly to sets of real numbers, is called the law of X (also just its distribution). It is the probabilistic fingerprint of X stripped of the experiment that produced it.

Informally, think of it as transferring probability through the function X. To find the law's value on a set A of numbers, you collect every outcome the experiment could give whose X-value falls in A, and you add up their probability: law of X on A equals P(X is in A) equals P of the set of outcomes that X maps into A. This is exactly the idea behind P(X = 8) for two dice being the probability of all pairs summing to 8. The law lives entirely on the real line and no longer mentions the original sample space.

Why bother naming this? Because it makes precise what we mean when we say two variables have the same distribution: their laws agree, as assignments of probability to subsets of the real line, even if their experiments differ. The cdf, pmf, and pdf are all just convenient ways of recording the same law. In more advanced language the law is the pushforward of the underlying probability measure by the function X (an induced measure on the reals); you do not need measure theory to use it, but that is what it formally is.