A set is considered countable if its elements can be matched one-to-one with the natural numbers 0, 1, 2, .... This means that you can list the elements in a sequence, even if the set is infinite, like the set of all integers or the set of rational numbers. In contrast, a set is uncountable if it cannot be matched with the natural numbers. An example of an uncountable set is the set of real numbers, which includes all the numbers on the number line. This means there are "more" real numbers than there are natural numbers, as shown by Cantor's diagonal argument.