a locally ringed space

Geometry on a space is governed by the functions you allow on it. A ringed space is the bare-bones way to record this: a topological space X together with a sheaf O_X of rings, where O_X(U) is meant to be 'the ring of admissible functions on the open set U'. But to do geometry you need more than functions — you need to know, at each point, which functions vanish there, so you can talk about the value of a function at a point and about a function being a unit (nonzero, hence invertible) near that point. A locally ringed space is a ringed space whose stalks are local rings, which is exactly the extra structure that lets 'the value at a point' make sense.

Precisely, a ringed space is a pair (X, O_X) with O_X a sheaf of commutative rings on X; its stalk O_{X,p} at a point p is the ring of germs of functions near p. It is a LOCALLY ringed space if every stalk O_{X,p} is a local ring — a ring with a unique maximal ideal m_p. The residue field k(p) = O_{X,p}/m_p is then the field in which 'the value f(p)' of a germ f lives: f(p) is the image of f in k(p), and m_p is precisely the germs vanishing at p. A morphism of locally ringed spaces is a continuous map f: X -> Y together with a map of sheaves of rings going the right way (pulling back functions on Y to functions on X), subject to the LOCAL condition that the induced map on stalks O_{Y,f(p)} -> O_{X,p} sends the maximal ideal into the maximal ideal — i.e. it pulls functions vanishing at f(p) back to functions vanishing at p. That last condition is what makes the algebra match the geometry.

This is the abstract frame that every geometric object fits into: smooth manifolds (with the sheaf of smooth functions), complex manifolds (holomorphic functions), and — the whole point of scheme theory — schemes (Spec R with its structure sheaf) are all locally ringed spaces, and an affine scheme is precisely a locally ringed space isomorphic to (Spec R, O_{Spec R}). A subtle but essential point: the 'local' in the morphism definition is not automatic. There are maps of ringed spaces that are not maps of locally ringed spaces, and dropping the local condition would allow geometrically absurd morphisms — for schemes, the maximal-ideal condition is exactly what forces a ring homomorphism R -> S to induce the right continuous map Spec S -> Spec R.