When to use $\in$ or $\subseteq$? If I have a family of $n$ sets like this $\mathcal{F}=\{\{S_1\}, \{S_2\}, \dotsc, \{S_n\}\}$. 
What is the right notation: for some $i\;\{S_i\}\in\mathcal{F}$ or $\{S_i\}\subseteq\mathcal{F}$?
In general, when to use $\in$ and when to use $\subseteq$?
 A: The second suggested notation is wrong, because $\{S_i\}$ is not a subset of $\mathcal{F}$.  To be a subset, each element of $\{S_i\}$ must be an element of $\mathcal{F}$, and that is not true: $S_i$ is not an element of $\mathcal{F}$. What is in $\mathcal{F}$ is $\{S_i\}$, which is the set containing $S_i$. 
The first suggested notation is correct.
$\in$ denotes element of, $\subseteq$ denotes subset of.
A: Given a set $X$ we write $A\in X$ when $A$ is an element of $X$. We write $A\subseteq X$ when every element of $A$ is an element of $X$. It's slightly more difficult to express $\in$ because that is taken to be the basic relation from which we define everything else. So we can really just say that $A\in X$ when $A$ is an element of $X$ (which is not much to say).
For example, $\{a\}\subseteq\{a,b\}$ because every element of $\{a\}$ is an element of $\{a,b\}$. On the other hand $a\in\{a\}$.
Note that it is perfectly possible for a set to be both an element and a subset of another set. Consider for example $A=\{a\}$ and $X=\{a,A\}=\{a,\{a\}\}$. We have that $A\in X$ and $A\subseteq X$.
In your case, $\{S_i\}$ is an element of $\cal F$. It might be a subset of $\cal F$ in the case where $S_i=\{S_j\}$ for some $i$ and $j$. But your question does not include sufficient information to decide about that.
