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A set is countably infinite if it has a bijection with ℕ, meaning its elements can be arranged in an infinite sequence. Countable unions of countable sets remain countable. These sets, despite being infinite, are 'smaller' than the continuum—there are discrete, enumerable objects within them.
Enumerate sets like ℤ, ℚ, and ℕ × ℕ explicitly to see why they are countable; contrast with the reals.
You already know about injections, surjections, and bijections, and you have a sense of cardinality — the size of a set. For finite sets, cardinality is just the element count. For infinite sets, size is measured by the existence of bijections: two sets have the same cardinality if and only if there is a perfect one-to-one pairing between them. A set is countably infinite if it has the same cardinality as ℕ — meaning its elements can be listed as an infinite sequence a₀, a₁, a₂, ... with every element appearing exactly once. "Countable" captures the idea that the set's elements can, in principle, be counted out one by one.
The surprising content of countability is how many familiar infinite sets turn out to be countable despite appearing much larger than ℕ. The integers ℤ seem twice as big — they extend in both directions — but the sequence 0, 1, −1, 2, −2, 3, −3, ... lists every integer exactly once, establishing a bijection with ℕ. The natural number pairs ℕ × ℕ seem like a two-dimensional infinity, but Cantor's diagonal enumeration — going along diagonals (0,0), (1,0), (0,1), (2,0), (1,1), (0,2), ... — lists every pair. The rationals ℚ can be arranged in a grid with numerator along one axis and denominator along the other, then diagonalized (skipping repeated fractions), showing ℚ is countable. In each case, the trick is finding a systematic path through all elements.
These examples reveal that countability is robust under the operations that naturally produce new sets: the union of countably many countable sets is countable, and the Cartesian product of finitely many countable sets is countable. These closure properties are why countable sets appear everywhere in mathematics — the integers, rationals, algebraic numbers, finite strings over a finite alphabet, and Turing machines are all countable.
The concept becomes powerful precisely at its boundary: what is not countable? The real numbers ℝ are not countable — this is Cantor's diagonalization theorem. The existence of an uncountable set separates countability from mere infinity. You have already enumerated all the "discrete" infinite structures; the continuum is something qualitatively larger. Countability is thus both a ceiling (nothing countable can enumerate the reals) and a floor (any infinite set you can systematically list is at most countable), making it the fundamental dividing line in the theory of infinite sets.