the support of a distribution

The support of a distribution is the set of values the random variable can actually take, the places where probability genuinely lives. Everywhere else the probability is zero. If you imagine the pmf or pdf as a landscape, the support is the footprint where the landscape rises above the flat ground; outside it, the function is identically zero and the variable never lands there.

Concretely, for a discrete variable the support is the list of values with positive mass: the support of a single die roll is {1, 2, 3, 4, 5, 6}, and of a Poisson count is {0, 1, 2, 3, ...}. For a continuous variable the support is the region where the density is positive: a uniform on [0, 2] has support [0, 2], an exponential has support [0, infinity), and a normal has support the entire real line. Knowing the support tells you the range of possible answers at a glance and the limits you must use when integrating a density.

Getting the support right prevents silly mistakes. If you forget that an exponential variable is never negative, you might integrate its density from minus infinity and get nonsense. The support also explains an apparent paradox of continuous variables: for a continuous distribution every single point inside the support still has probability zero, because probability is area, not the mere fact of being a possible value. So a value can be in the support (positive density there) yet have probability exactly zero of occurring on the nose.