What is the difference between "family" and "set"? What is the difference between "family" and "set"?
The definition of "family" on mathworld (http://mathworld.wolfram.com/Family.html) is a collection of objects of the form $\{a_i\}_{i \in I}$, where $I$ is an index set. But, I think a set can also be represented in this form. So, what is the difference between the concept family and the concept set? Is there any example of a collection of objects that is a family, but not a set, or reversely?
Many thanks!

A family is indeed a set, and it is defined by the indexing -- as you observed.
Just as well every set $A$ is a family of the form $\{i\}_{i\in A}$.
However often you want to have some property about the index set (i.e. some order relation, or some other structure) that you do not require from a general set. This addition structure on the index can help you define further properties about the family, or prove things using the properties of the family (its elements are disjoint, co-prime, increasing in some order, every two elements have a supremum, and so on).
