From Spivak's Category Theory for the Sciences:
Definition 6.1.2.1 (Diagrams). Let $C$ and $I$ be categories. An $I$-shaped diagram in $C$ is simply a functor $d:I \rightarrow C$. In this case $I$ is called the indexing category for the diagram.
The example which followed was (for me) unpleasant to read and did not make the concept stick:
I brushed off knowing rigorously the definition of $I$, but the idea is aggressively repeated.
... Repeated too aggressively to ignore.
Can someone give me an example (with an illustrated diagram) that is as primitive and unpretentious as possible? Have it be as plainly-worded and intuitive as the integer or natural number examples you see in good textbooks.
If you take $I$ to be a discrete category, i.e. a category without any non-identity morphisms, then this will just be an indexing set. We have no non-trivial composable morphisms to worry about and just map identity morphisms to identity morphisms. Let us make this concrete in an example:
Let $I = \mathbb{N}$ considered as a discrete category. To define a functor $I \rightarrow C$, we just need to choose an object $X(n)$ for every natural number $n$. So an $I$-shaped diagram in $C$ is just a sequence of objects in $C$ indexed by $I = \mathbb{N}$. The same is true for any discrete index category.
In general you should think about these things as follows:
The index category will be an "abstract diagram", i.e. just some shape like
$$\bullet \rightarrow \bullet \rightarrow \bullet,$$
where I do not draw the identity morphisms and the composition for simplicity. Defining a functor $I \rightarrow C$ now boils down to choosing exactly such a shape within the given category $C$. In this case, we need to choose three objects $X,Y,Z \in C$ for the three bullets together with two morphisms $X\rightarrow Y$ and $Y \rightarrow Z$. So in total our functor maps $$\bullet \rightarrow \bullet \rightarrow \bullet \hspace{15pt} \text{ to } \hspace{15pt} X \rightarrow Y \rightarrow Z.$$
To summarize: The index category is an abstract "nameless" diagram with no connection to the given category $C$ and the functor (i.e. the diagram) is a choice of "names" for the abstract diagram inside of $C$. In the example above I decided to let the first bullet point be my object called $X$ etc.