What is the difference between the relations $\in$ and $\subseteq$? What is the difference between the relations "$\in$" and "$\subseteq$" ? Don't they both mean that something is an element of a set? Are they interchangeable in some or all situations?

Like: 

$x \in A$ ($X$ is an element of the set $A, X$ is in $A, A$ contains $X$)

$x \subseteq A$ ($X$ is an element of the set $A, X$ is in $A, A$ contains $X$)
The following are true:

$$rock\in \{rock,paper,scissors\}$$
$$\{rock\} \subseteq\{rock,paper,scissors\}$$
$$rock\not\subseteq \{rock,paper,scissors\}$$
$$\{rock\}\not\in\{rock,paper,scissors\}$$

editted to make it clearer.
As I said in comment $x\in A$ is about membership of the element $x$ in $A$. But $X \subseteq A$ is about inclusion of the subset $X$ of $A$ in $A$. see also here and here.