Schemes are objects central to the study of algebraic geometry, but what are they? A scheme is a locally ringed space that admits an open cover by affine schemes, sure, but what is a locally ringed space? What is an affine scheme? I will try to answer these questions by motivating the definition of a scheme with basic real and complex analysis. The basic yoga is that schemes are weird, because commutative rings are weird; any non-abelian category is weird. But Riemann Surfaces are not weird at all, and much intuition for how schemes behave, and what sorts of things should be true about them, can be obtained through the lens of complex geometry where one has holomorphic structure to work with - contractible neighborhoods, differentiability, and other such familiar and friendly notions. Once we have drawn some pictures and seen, at least heuristically, why we should think of commutative rings as the natural dwelling place for functions, we will define abstractly what a scheme is and give some examples.
What Is a Scheme?