Difference between $\vec{a} $ and $||\vec{a}||$ What exactly is the difference between $\vec{a} $ and $\lVert\vec{a}\rVert$ ?
I know  $\lVert\vec{a}\rVert$ is the magnitude of a vector $\vec{a} $ , but what is $\vec{a} $  then exactly? I know it's a vector, but how can you represent it? Let's say $\vec{a} $  is a vector in $\mathbb{R}^2$, if you draw $\vec{a} $ in $\mathbb{R}^2$, don't you automatically draw $\lVert\vec{a}\rVert$ ?
Also, what is the difference between $ \lVert\vec{a}\rVert \cdot \big\lVert\vec{b}\big\rVert $ and  $\vec{a} \cdot \vec{b} $?
To me it seems like multiplying the same, but I know it isn't.   

$\vec{a}$ is a vector, i.e. some element $(a_1,a_2)$ of $\mathbb{R}^2$ in your case. You draw it by drawing an arrow from the point $(0,0)$ to the point $(a_1,a_2)$.
$||\vec{a}||$ is the length of the vector $\vec{a}$, i.e. the length of the arrow from $(0,0)$ to $(a_1,a_2)$. $||\vec{a}||$ is thus a positive real number, i.e. it is an element of $\mathbb{R}$ (and is $\geq 0$), not of $\mathbb{R}^2$.
$||\vec{a}||\cdot||\vec{b}||$ is the product of the lengths of $\vec{a}$ and $\vec{b}$. It is thus also a positive real number. For a vector $\vec{a} = (a_1,a_2)$ the length $||\vec{a}||$ is $\sqrt{a_1^2 + a_2^2}$.
$\vec{a}\vec{b}$, sometimes also written $\left<\vec{a},\vec{b}\right>$ is the scalar product of $\vec{a}$ and $\vec{b}$. It's also an element of $\mathbb{R}$, not of $\mathbb{R}^2$. For two vectors $\vec{a} = (a_1,a_2)$ and $\vec{b} = (b_1,b_2)$ the definition of the scalar product is $\vec{a}\vec{b} = a_1b_1 + a_2b_2$.
Note that there is a connection between the two. By the definition of the scalar product above, $\vec{a}\vec{a}$ is $a_1^2 + a_2^2$. This is the same as $||\vec{a}||^2$.
