A lot of it comes down to set theory and the idea of mappings between sets. So countably infinite sets are those for which there exists a bijective (one-to-one and onto) mapping f:ℤ→X which is an extension of the idea of counting a finite set by showing that there’s a mapping of the set {1, 2, …, n}→X for a set X of size n. From this, we’re able to find larger infinities like |ℝ| where there is no bijective map from ℤ to ℝ. David Foster Wallace (yes, that David Foster Wallace) wrote a reasonably readable book about infinity that could be a good starting point in exploring the subject.