**Definition.** [Hartshorne] If $X$ is any scheme over $Y$, an invertible sheaf $\mathcal{L}$ is very ample relative to $Y$, if there is an imersion $i\colon X \to \mathbb{P}_Y^r$ for some $r$ such that $i^\ast(\mathcal{O}(1)) \simeq \mathcal{L}$.

My question is: what is the right way (interpret "right way" as you wish) to think about very ample sheaves? In particular, why is the word "ample" being used? What is it that I have an ample amount of? Degree 1 elements?

In the simple case when $Y=\text{Spec}(A)$ is affine then $i^\ast(\mathcal{O}(1))$ is just $\mathcal{O}(1)$ as defined on $\text{Proj} A[x_0,\ldots x_r]$, that is, it's the sheafification of the degree 1 part of the polynomial ring $A[x_0,\ldots,x_n]$. So it seems like the more general definition is just meant to generalize this phenomenon. Is this true? If so, why is it worth generalizing? What's special about degree 1 elements? The only thing I can think of is that the polynomial ring is generated as an $A$-algebra by its degree 1 elements.

As you can tell, my question is not very well formed, so feel free to add anything you think is relevant. I am also happy to expand on anything I've written here.