Are there multiple types of "components" in the study of vectors? In class we were taught how to write any vector in "component form" as $(a, b)$, where $a$ is "change in x" and $b$ is "change in y." 
However, yesterday we were also taught how to "decompose" a vector into its "components," involving methods like projection:

What, if anything, is the relationship between these two apparently different definitions of a "vector component"? Or is there in fact no difference? I'm confused why the terminology is used in both contexts, and in turn I think that means I don't entirely understand the underlying concepts themselves. 
 A: In general, the "component" of a vector in a certain direction is to what extent that vector is pointing in that direction.
So the usual notation $\vec{v}=(x,y)$ says the vector is pointing "$x$ units" in the direction of the $x$-axis, and "$y$ units" in the direction of the $y$-axis.  Notice that $x$ and $y$ are also the projection of $\vec{v}$ onto the usual axes.  So $\vec{v}=(proj_{(1,0)}(\vec{v}) \, , \,proj_{(0,1)}(\vec{v}))$
Similarly in your examples, the component of $\vec{v}$ in the direction of some vector $\vec{u}$ is the projection $proj_{\vec{u}}(\vec{v})$ 
A: It is actually the same thing. 
In the first usage of the word, you find that the components you get are exactly equal to the $x$ and $y$ co-ordinates of the vector $u$. You have found the components of $u$ in the $x$ direction and in the $y$ direction, but to save breath, we don't say this in full.
In the second usage, it is perhaps not intuitively clear how the $x$ and $y$ components of $u$ combine to give the $v$ component of $u$. The dot product formula gives this. In fact, if you changed your co-ordinate system from units of $x$ and $y$ into units of $v$ and $v_\perp$, where $v_{\perp}$ is perpendicular to $v$, then you'd get $$u=av+bv_\perp\tag1$$ for some numbers $a,b$. Then $a$ is the component of $u$ in the direction of $v$. In this new co-ordinate system, (called a basis), one can write $u=(a,b)$ as a short-hand for the expression $(1)$. Then, this looks exactly like the first usage of the word components.
