In algebraic geometry one (mostly) studies varieties given by polynomial equations.
Such equations define algebraic varieties and there are many "dictionaries" available.
For example, the category of rings is anti-equivalent to the category of affine schemes, etc.
What if we enlarge our realm of possible equations to equations of the form
$x^y+y^z = z^x$ over the rational numbers.
These also define "varieties" which are no longer algebraic. It doesn't define a scheme, I think, but can we define a locally ringed space or something similar or maybe more general to it?
Sorry for the vagueness. I just have the feeling that sometimes one should enlarge their category in order to get their hands on what's really going on.
It sounds like you're talking about a complex analytic space, sometimes called an analytic variety. It is locally defined by the vanishing of a holomorphic function in some $\mathbb C^n$. For instance $\mathbb Z$ is not an algebraic subvariety of $\mathbb C$ but it is an analytic subvariety because it is the zero set of $\sin \pi z$.