I'm studying some representation theory on $S_n$ and $GL(V)$ and tensor spaces, and have come across a lot of material involving rational representations. I'm not really an algebraic geometer by trade, though I have covered some basics before, and I am essentially looking for some intuition behind what a rational map actually is, beyond the algebraic geometry definition. In my opinion, the (formal) definition is as ugly as sin, and any "informal" definition which I sometimes find accompanying the formal one is almost equally dire.
I feel like, given its definition as a "partial function between algebraic varieties", and also its name, it is in some manner an extension of the idea of a quotient of 2 polynomial functions $f(X)/g(X)$ (which would of course be defined everywhere where $g$ is nonzero). However, none of the materials I can find online are proving very fruitful in developing an understanding of what these maps are really meant to "be", beyond the raw definition.
I would just like to hear, in simple terms if possible, what you understand a rational map to be, if there is actually more to be said than just the definition: for example, is there some result which says "on nice (affine, irreducible, etc?) spaces, a rational map is of the form $f/g$, polynomials $f$, $g$" as above, or something along those lines? Or, are there just certain spaces (I normally only need to worry about affine things like $\mathbb{C}^n$, for example!), where a rational map always has some nice form? I should add, I don't particularly have any reason to think a rational map should be some quotient $f/g$ of polynomials, except for the terminology and the fact that the definition seems to permit such maps. If I am in totally the wrong ballpark, please let me know as well, I am simply seeking a better understanding of what the definition is really saying, even if there are cases where such maps always look nice and cases where they don't.