The strict mathematical definition of a countable set is a set that is in bijective correspondance to the natural numbers, which are the integers from. Basically, this means that you can assign a natural number to every element in the set, so in essence you are "counting" the set even though it is infinite. For example, the rational numbers are a countable set since you can write a pattern which will generate all rational numbers, and then just assign the natural numbers to this pattern in order.
Countably infinite sets are the "smallest" infinite sets, there are also uncountable infinite sets such as the real numbers or complex numbers, in which it is impossible to write a pattern which will explicitly write all of the reals. So these are larger infinities.