The cartesian product of two sets : $X,Y$ is a set $Z$ defined as :
$Z = \{ (x,y) \, | \, x \in X \, \text {and} \, y \in Y \}$
where $(x,y)$ is the ordered pair having $x$ as first component and $y$ as second component.
Thus, the cartesian product $X \times Y$ is the set of all ordered pairs with first component in $X$ and second component in $Y$.
A relation $R$ with domain in $X$ and range in $Y$ is a subset of the cartesian product $X \times Y$, i.e. :
$R \subseteq X \times Y$.
Thus, a relation is a set of ordered pairs.
A function $F$ is a relation satisfying the ("functionality") condition :
if $(x_1,y_1) \in F$ and $(x_1,y_2) \in F$, then $y_1=y_2$.
A binary operation $f : Y \times Y \to Y$ is a function from the cartesian product $Y \times Y$ to the set Y, i.e. a subset of $(Y \times Y) \times Y$, because it "maps" an ordered pair $(y_1,y_2)$ into an element $y_3$, with $y_i \in Y$.
You can try to clarify the definitions with some simple examples.
Let $\mathbb N = \{ 0, 1, 2, ... \}$ the set of natural numbers.
Consider the cartesian product $\mathbb N \times \mathbb N$ and :
the relation $<$ ("Less then"), i.e. $(n,m) \in L$ iff $n < m$,
the function $s$ ("Successor"), i.e. $(n,m) \in S$ iff $m=s(n)$
the (binary) operation $+$ ("Plus"), i.e. $((x,y),z) \in P$ iff $z=x+y$.
A general element of $X_1\times X_2$ is an ordered pair $(x_1,x_2)$, whereas a general element of $\prod_{i=1}^2 X_i$ is a function from $\{1,2\}$ to $X_1\cup X_2$ where $f(1)\in X_1$ and $f(2)\in X_2$. That is, the former contains
$$
(x_1,x_2)
$$
while the latter contains
$$
\{(1,x_1),(2,x_2)\}
$$
Therefore, the elements of $X_1\times X_2$ and $\prod_{i=1}^2 X_i$ are "cosmetically" different, but there is a bijective correspondence between these two sets, where one of the two above objects is mapped to the other. This is the reason that the point being made should be "promptly forgotten;" the two constructions capture the same concept in different ways, so can be used interchangeably in any application.
It should be noted, then promptly forgotten, that the ordered pair notation $(a,b)$ is actually shorthand for a particular set, usually chosen to be $\{\{a\},\{a,b\}\}$.
Best Answer
The set $\{x\}\times \{x\}$ is the singleton containing the ordered pair $(x,x)$, i.e. $\{x\}\times\{x\} = \{(x,x)\}$. By definition, $(x,x) = \{\{x\},\{x,x\}\}$. This last set is equal to $\{\{x\},\{x\}\}$ since $\{x,x\} = \{x\}$, and this final set is equal to $\{\{x\}\}$ since $\{x\} = \{x\}$. Hence $\{x\}\times\{x\} = \{\{\{x\}\}\}$.