What makes a random variable discrete?
This just means that we can LIST the values it takes.
For instance, the set of values a fair die can take may be listed as $1,2,3,4,5,6$.
The list may be infinitely long, such as $0,1,2,3, \dots $
The set of values a discrete random variable can take is its “support”.
In technical work, we treat discrete random variables quite differently to other types.
It is possible to formalise the notion of “listing” the values, in terms of the mathematical idea of a “countable set”.