So, the cardinality of a finite countable set is the number of elements in the set. Some examples of such sets are N, Z, and Q ( rational numbers). If a set is countable and infinite then it is called a 'countably infinite set'.
(This point is used to determine whether an infinite set is countable.) To be precise a set A is called countable if one of the following conditions is satisfied. On the other hand, the set of all real numbers R is uncountable as we cannot list its elements and hence there can't be a bijection from R to N. In other words, we can have a one-to-one correspondence ( bijection) from each of these sets to the set of natural numbers N, and hence they are countable. For example, sets like N ( natural numbers) and Z ( integers) are countable though they are infinite because it is possible to list them. Yes, finite and infinite sets don't mean that countable and uncountable. An infinite set, on the other hand, has an infinite number of elements, and an infinite set may be countable or uncountable. A finite set is a set with a finite number of elements and is countable.
