Trouble understanding what a natural transformation is I understand that an arrow is between two objects, a functor is between two categories. And then a natural transformation is, according to Goldblatt's Topoi, a comparison of two functors.
Here is Goldblatt's definition:

A natural transformation from functor $F: \mathscr C \to \mathscr D$ to functor $G: \mathscr C \to \mathscr D$ is an assignment $\tau$ that provides, for each $\mathscr C$-object $a$, a $\mathscr D$-arrow $\tau_a :F(a) \to G(a)$, such that for any $\mathscr C$-arrow $f:a\to b$, the diagram commutes in $\mathscr D$, ie. $\tau_b \circ F(f)=G(f)\circ \tau_a$. We use the symbolism $\tau: F \xrightarrow{\tau} G$ to denote that $\tau$ is a natural transformation from $F$ to $G$.


I have absolutely no idea what's going on. I tried to understand the diagram, but I can't even understand where the $F(a)/F(b)$ and $F(f)$ are coming from. (Is this similar to how structures are preserved under homomorphism? Because the way the composition is formulated looks a bit similar)
I think I am lacking an intuitive understanding of what it means to compare two functors.
Could anyone help please?

It's been pointed out that another question has already covered what I am asking (What intuitive notion is formalized by "natural transformation" in category theory?); the question does also ask for an intuitive perspective on natural transformation.
However it's framed around topology and other mathematical structures, but I really would like to avoid those because coming from a non-mathematical background, framing it around those does not really help (I have no idea what those structures are). If possible, I would like to keep any other structure away from the explanation.
 A: A natural transformation is between two functors, just as a functor is between two categories, i.e. they are simply arrows in a suitable category. The word 'comparison' has no special meaning here.
Note that in Goldblatt's definition, $F$ and $G$ are given functors, both $\mathscr C\to\mathscr D$, so that they already assign the object $F(a)$ [resp. $G(a)$] of $\mathscr D$ to any object $a$ of $\mathscr C$. 
Also, they assign arrows $\underset{a\to b}f\ \mapsto \underset{F(a)\to F(b)}{F(f)}$ [resp. $\underset{G(a)\to G(b)}{G(f)}$].
We can also view natural transformations simply as functors $\mathscr C\to(\mathscr D^\to)$ where $\mathscr D^\to$ is the arrow category of $\mathscr D$, whose objects are the arrows of $\mathscr D$, and a morphism $\underset{a\to b}f\ \to \ \underset{a'\to b'}g$ is defined to be a commutative square
$$\matrix{a&\overset f\to & b\\ \downarrow & & \downarrow \\ a' & \underset g\to & b'}$$
In that setup, i.e. if a functor $\varphi:\mathscr C\to (\mathscr D^\to)$ is given, we can recover $F$ and $G$ by composing $\varphi$ with the domain [resp. the codomain] functor $(\mathscr D^\to) \to\mathscr D$, which assigns its arrow on the left [resp. right]  to a commutative square. 
