What is the value of the intersection of X and the set containing X? How to calculate X $\cap$ $\{X\}$ for finite sets to develop an intuition for intersections?
If $X$ = $\{$1,2,3$\}$, then what is $X$ $\cap$ $\{X\}$? 

$\{X\}$ contains one element and one element only.  So as $E \cap F \subset F$ we know $E \cap \{X\} \subset \{X\}$.  So either $E \cap \{X\} = \{X\}$ if $X \in E$ or $E \cap \{X\} = \emptyset$ if $X \not \in E$.  
It violates the axioms of set theory to have a set such that $X \in X$ (a set can't be an element of itself). So $X \in X$ so $X \cap \{X\} = \emptyset$.
As per your example.
$\{1,2,3\} \cap \{\{1,2,3\}\}$....  $1 \not \in  \{\{1,2,3\}\}$, $2 \not \in  \{\{1,2,3\}\}$, $3 \not \in  \{\{1,2,3\}\}$, and $\{1,2,3\} \not \in \{1,2,3,\}$.  And $1,2,3,\{1,2,3\}$ are everything in either set and none of them  are in both sets. so $\{1,2,3\} \cap \{\{1,2,3\}\} = \emptyset$.
Note:  It doesn't matter if $X$ is finite or infinite or empty.  $X \cap \{X\} = \emptyset$.  (Unless you ignore the axiom that $A \not \in A$.)
