But I think the way it’s usually phrased is, a “countable set” is one that can be put in one-to-one correspondence with the natural numbers.

So the set of natural numbers is countable, simply by corresponding each number with itself. And the set of integers is countable, by associating 0 (integer) with 0 (natural number), 1 with 1, -1 with 2, 2 with 3, -2 with 4, etc. There’s even a way of showing that the rational numbers (i.e. fractions) are countable, too.

Some sets (real numbers, for instance) aren’t countable — no matter how you put them in correspondence with the natural numbers, some will ‘fall through the cracks.’

But infinity is a different thing — it, like a natural number, is a ’size of a set.’ The natural numbers are ‘countably infinite.’ The real numbers are ‘infinite, and not countable.’

When people talk about ‘different infinities,’ I believe they’re talking about these sorts of ‘levels’ of uncountability. You can, axiomatically, give ’sizes’ to different infinite sets, based on how they correspond to the natural numbers. If you’re countably infinite, you are given the size of “infinity-0″. And if you can correspond, in a one-to-one way, with sequences (I think) of natural numbers, you’re “infinite-1″. And so on.

Anyway. Like I said, I am not a mathematician.