Im reading a book and somewhere it says:
A partition of a set $A$ is a family of subsets of $A$, $\{A_a\}_{a\in I}$ such that
Where $I=\{x\mid x=1,\ldots,n\}$.
$A_a\neq A_b\implies A_a\cap A_b=\emptyset$
...
What does it mean? Does family just mean set? Is $\{A_1,\ldots,A_n\}$, $\{\{A_1,\ldots,A_n\}\}$ or $\{\{A_1\},\ldots,\{A_n\}\}$ a partition of $A$?
On
A partition of a set $A$ is simply a collection/set of disjoint subsets (i.e. for any pair of subsets in the collection, their intersection is zero) of $A$ such that their union is $A$.
For example, let $P$ be partition of $A$ into $n$ parts, then $P=\{A_1,A_2,...A_n\}$ where $A_1\cup A_2\cup...\cup A_n = A$ and $A_i\cap A_j = \emptyset, \forall A_i\neq A_j$.
Depending upon the usage, $P$ may or may not be allowed to contain the empty set. The former case might be referred to as a weak partition.
Regarding your confusion:
A partition of a set $A$ is a family of subsets of $A$, $\{A_a\}_{a\in I}$ such that
Where $I=\{x\mid x=1,\ldots,n\}$.
This means that the partition is made up of $n$ subsets of $A$, denoted by $\{A_a| \forall a \in \{1,2,...n\}$. This is the same as the definition given above, just that it is in different("set-builder") notation.
A family of subsets of $A$ just means a set whose elements are all subsets of $A$. So, a partition would look something like $\{A_1, ..., A_n\}$, where each $A_i \subseteq A$.
Note that there need not only be finitely many sets in the family (although in your case, there is). In that case, we introduce the notion of an indexing set. In the example above, our indexing set would be $\Bbb{N}_n$ (the set of all natural numbers less than or equal to $n$); this is to say, for each $i\in \Bbb{N}_n$, there is a set $A_i$ in the family.