Any set which can be put in a one-to-one correspondence with the natural numbers (or integers) so that a prescription can be given for identifying its members one at a time is called a countably infinite (or denumerably infinite) set.

What do you mean by uncountably infinite set give an example?

Uncountable is in contrast to countably infinite or countable. For example, the set of real numbers in the interval [0,1] is uncountable. There are a continuum of numbers in that interval, and that is too many to be put in a one-to-one correspondence with the natural numbers.

Is Z countably infinite?

The set Z of integers is countably infinite.