Family and the role and nature of the 'index'

Jan 2010
Hobart, Tasmania, Australia
I am seekinhg to understand example 4 on page 9 of Cohn's book "Introduction to Ring Theory"

The example needs a clear understanding of the nottion of a family which Cohn briefly introduces on page 3 where he defines the notion of a set and members of a set and then defines a family mentioning how families are indexed (see attachment showing Cohn page 3) .

I am struggling with this notion and would appreciate someone giving me an example which clearly shows the nature and role of the index.