What does discrete mean?
If the set of values assigned to the random variable is countable, then we call it a discrete random variable.
A countable set is either a finite set or an infinite set.
Hence a discrete random variable can be assigned a finite set of values or an infinite set of values.
The important point here is that the set of values so assigned must be countable.
In other words, we should be able to identify which is the first value, which is the second value and so on.
for a continuous random variable, the set of values is uncountable. In other words, we can never identify which is the second value, which is the third value etc.
.>For example consider the random variable whose values are in the range 0 < X < 1.
Here we don't know which is the first value or which is the second value etc. Such a set is called uncountable.