When looking at mathematical definitions, there are quite a few cases where we limit certain properties to countably infinite sets (e.g. $\sigma$-Algebras).
In some cases we set this limit as we'd lose our intuition of what would happen if we chose an even bigger infinity, in other cases there are hard facts at work.
Yet, the only property of countably infinite sets that comes to my mind is ... well, that they are countable, and the higher cardinalities are not.
I'm sure though that there are many more characteristic properties of countably infinite sets that get lost when moving up to higher cardinalities - so what are they?