What does the dot product of two vectors represent? I know how to calculate the dot product of two vectors alright. However, it is not clear to me what, exactly, does the dot product represent.
The product of two numbers, $2$ and $3$, we say that it is $2$ added to itself $3$ times or something like that.
But when it comes to vectors $\vec{a} \cdot \vec{b}$, I'm not sure what to say. "It is $\vec{a}$ added to itself $\vec{b}$ times" which doesn't make much sense to me.

First of all, if we write $\vec{a} = a \vec{u}$ and $\vec{b} = b \vec{v}$,
where $a$ and $b$ are the length of $\vec{a}$ and $\vec{b}$ respectively,
then $$\vec{a} \cdot \vec{b} = (a \vec{u})\cdot (b \vec{v}) 
= ab \,\, \vec{u} \cdot \vec{v};$$ 
this is a pretty natural
property for a product to have.
Now as for $\vec{u} \cdot \vec{v}$, this is equal to $\cos \theta,$
where $\theta$ is the angle between $\vec{u}$ and $\vec{v}$.
As King Squirrel notes, this is also the length of the projection of $\vec{u}$ onto the line through $\vec{v}$, and also the length of the projection of $\vec{v}$ onto the line through $\vec{u}$.
So altogether we get 
$$\vec{a} \cdot \vec{b} = a b \, \cos \theta,$$
and it has the interpretation in terms of projecting one vector onto another
that King Squirrel discusses.
