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' can have the pedestrian meaning of 'a set of sets'. So instead of confusing the reader with 'set' too many times, you can say 'family of sets' instead'. For example, a hypergraph (or really the edges of a hypergraph) is a 'family of sets of vertices'.
A: As @lhf says, a family is a function $I\to U$. While it is true that every set can be though of as a family indexed by itself, not every family is of this form. For example, a single element of $U$ may occur more than once in a family (with different indices).
A: Sometime one encounters a phrase like "family of sets with property X".
In this case the use of "family" as a mathematical object is more or less informal and not well defined (within ZFC), since such thing isn't necessarily a set. You should read "for each member of the family ..." as an abbreviation of "for each set that has property X ..." and try not to think of it as a real set too much. If you do, you might run into trouble sooner or later. For instance if you start wondering whether the family of all non-empty sets contains itself -- this is all related to Russel's paradox of course.
(There is a way to get rid of this problem by introducing classes into your set theory.)
A: Strictly speaking, a family is a function $I \to U$, where $I$ is an index set and $U$ is a universe that contains the members of the family.
Strictly speaking, a set is not a family indexed by itself: it's either the image of the family, if the members are the elements, or the union of that family, $\cup_{x\in X} \{x\}$, if the members are singletons.
A: Family is another way to express function.
If $X$ and $Y$ are sets, a function from $X$ to $Y$ is a relation $f$ such that $\operatorname{dom} f = X$ and for each $x \in X$ there is a unique element $y$ in $Y$ with $(x, y) \in f$. 
$X$ is index set, the range of the function is indexed set. The function $f$ is called family.
A relation is a set of ordered pairs, ordered pair, eg: $(a,b)$, defined by $\big\{\{a\}, \{a,b\}\big\}$, so $a$ is the first coordinate, $b$ is the second.
Quote from the mathworld link:

Every set $X$ gives rise to a family
$f:X\to X,\, f(x)=x$

A: 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).
