A finite set is like a package for a group of things. The package can be monstrously big to fit the amount of things it needs to hold but ultimately there is a limit to the amount of items in the package. This means that if you were to go by each item in this package one by one and count them you would eventually reach the last item. (Incidentally this number that you counted would be the cardinality of the set)
More formally a Finite set is a set that also has a bijection with one of the natural numbers.