how do you know if a system is a group?

Question

for example, what four things make something like (Z8, +) a group
. the 8 is supposed to in subscript.

Answer 1

You need a set together with an operation on the set. In your example, the set is Z_8, the operation is +. Then there are three axioms which must be satisfied:

(1) the operation is associative,
(2) there is an identuty element,
(3) every element has an inverse.

If you don't know what those terms mean, read your textbook.