The usual way I used to imagine Differentiable Manifolds are, as locally Euclidean spaces, with some nice enough properties. Intuitively speaking, instead of just looking at the manifolds from 'outside' (as embedded in some Euclidean space), I sometimes try to think of manifolds from the 'inside' , ie, I try to imagine how it looks to an observer living inside the manifold, atleast in the 3-dimensional case. Thinking in this way, locally Euclideanness becomes not just a tool to calculate co-ordinates around a point, but the exact way how our universe looks to us, the observers inside it.
These point of view has motivated me so much, that I was wondering if there was a similar kind of geometric visualisation possible for algebraic varieties. That is, is there some general property it follows when an 'observer' looks at it from the 'inside'?
Sorry if this question is too vague/intuitive.