what is the property that distinguishes finite sets from infinite sets (give examples of each to accompany explaination).

Question

finite sets are countable. Infinite sets are not.

so what would be an example of an infinite set? one that never ends?

yes. 1,2,3,4,... is an infinite set.
1,2,3,4,5,6 is a finite set.

Countable can also refer to infinite sets that can be put into one to one correspondence with the integers.

E.g. the set of all fractions is countable, while the set of reals is not countable.

Answer 1

You are right