I.e.:
Identity For any element a in G there is an element e where e⋆a = a⋆e = a.
Invertibility For each element a in G there is an element b where a⋆b = b⋆a = e, where e is the identity.
Invertibility: given any a, there is b such that a⋆b = b⋆a = e. For two different a's you get two different b's.
As example let G be a set of all strings and the operation is appending. There is element e == "". For any string a it holds that if you append empty string to any side, you get the original string back. However there is no inverse. You can not append two string to form an empty string. Apparently, strings with appending do not form a group.
Group example: let G be a set of all integers and the operation is addition. Then e is 0, as for all a it holds that a + 0 = 0 + a = a, and for each a you can find -a such that a + (-a) = (-a) + a = 0.
If you want an example, think Z (integers) and "+": there is an element of Z, namely zero (0), where the identity rule tells us that x + 0 = 0 + x = x and the invertibility rule says that for any x in Z, there is an y in z such that x + y = 0
Invertibility is existence of Yin for a Yang, superhero for a supervillain, north pole for a south and combining them creates a neutral.