If $A = \{a,b,c,d\}$, what does $\{\{a\}\}$ mean? I am trying to understand the significance of curly brackets in set theory.
Let $A = \{a,b,c,d\}$.
I understand $A$ is a set that includes the objects $a,b,c,d$.  However, what does $\{\{a\}\}$ mean?  Why use two curly brackets on either side instead of just $\{a\}$ ?
 A: $\{\{a\}\}$ means the set containing the set containing $a$.
Think of sets like bags.  $a$ is an apple.  $\{a\}$ is an apple in a bag.  $\{\{a\}\}$ is an apple in a bag, that is in another bag.
A: How does $\{a\}$ differ from $\{\{a\}\}$? Well, Omnomnomnom and vadim123 have pointed out the key differences, but perhaps a small elaboration will help. 
Is $a\in\{a\}$? Yes. Is $a\in\{\{a\}\}$? Well, $a\neq\{a\}$ so the answer is a resounding no. Ah, but do you know what it means for a set to be a subset of another set? If $A$ is a subset of $B$, then this is often denoted by $A\subseteq B$ and may be thought of as "if $x$ is $A$, then $x$ is in $B$"; symbolically, $A\subseteq B$ may be represented as $x\in A\to x\in B$. This illustrates some key points:


*

*$a\in \{a\}$

*$\{a\}\in\{\{a\}\}$

*$\color{blue}{\{a\}\subseteq \{a\}}$

*$\color{blue}{\{a\}\not\in\{a\}}$


Pay special attention to the last two points highlighted in blue. 
A: The set $\{\{a\}\}$ is the set whose only object is the set $\{a\}$.  $\{a\}$, in turn, is a set whose only object is $a$.
It might help to consider the set 
$$
\{a,\{a\},\{b\}\}
$$
whose objects are $a$, the set $\{a\}$, and the set $\{b\}$.  Note that this set is quite different from $\{a,b\}$.
A: If $\{a\}$ is the set whose only element is $a$; denote that set by $x$, then $\{x\}$ is the set whose unique element is $x$, so its unique element is $\{a\}$, which in other words mean that it is the set $\{\{a\}\}$.
