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.
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.