Reduced, irreducible, integral, separated, proper — the adjectives of scheme theory
Schemes come in many flavors, and a large part of algebraic geometry consists of identifying the right adjectives — properties that constrain a scheme's behavior in useful ways. Some properties, like being reduced (no nilpotents) or irreducible (not a union of two proper closed subsets), describe the local algebra. Others, like being separated or proper, are global conditions that mimic Hausdorffness and compactness from topology.
A scheme that is both reduced and irreducible is called integral — its structure sheaf consists of integral domains. When it is also Noetherian (every open affine is Spec of a Noetherian ring), the standard tools of commutative algebra apply. At the smoothest end of the spectrum, a smooth scheme has local rings that are regular — the algebraic analog of a manifold with no singularities.
A scheme X is reduced if 𝒪X(U) has no nonzero nilpotent elements for every open U — equivalently, every stalk is a reduced local ring. It is irreducible if the underlying topological space cannot be written as a union of two proper closed subsets, which means it has a unique generic point. An integral scheme is both reduced and irreducible; its function field K(X) is the stalk at the generic point.
The Noetherian condition — every ascending chain of open subsets stabilizes — ensures that the scheme has finitely many irreducible components and that ideals in the structure sheaf are well-behaved. Most schemes encountered in practice (varieties, arithmetic schemes, moduli spaces) are Noetherian.
A morphism f: X → S is separated if the diagonal Δ: X → X ×S X is a closed immersion. This is the algebraic analog of the Hausdorff condition: it ensures that limits of sequences are unique and that the intersection of two affine open subsets is again affine. Most reasonable schemes (and all affine schemes) are separated.
A morphism is proper if it is separated, of finite type, and universally closed (the image of every closed set remains closed after any base change). This is the algebraic version of compactness — the valuative criterion says a morphism is proper if and only if every "arc" Spec(k[[t]]) → X over S can be uniquely completed. A scheme is smooth over a field k if all its local rings are regular, meaning the tangent space has the expected dimension everywhere.