The cardinality of a set describes how many elements it contains. For finite sets, this is ordinary counting. For infinite sets, size is compared through functions.

Same cardinality

Sets $A$ and $B$ have the same cardinality when there is a bijection

$$f:A\to B.$$

A bijection pairs every element of one set with exactly one element of the other.

Infinite sets behave differently

The natural numbers and the even natural numbers have the same cardinality because

$$f(n)=2n$$

is a bijection between them, even though the even numbers form a proper subset of the natural numbers.

This cannot happen for finite sets.

Comparing sizes

An injection from $A$ into $B$ shows that $A$ is no larger than $B$ in cardinality. A bijection shows equality of cardinality.

The idea of cardinality therefore extends counting beyond finite collections without requiring infinite sets to behave like very large finite sets.