nLab
integral scheme

An algebraic scheme $X$ is integral if for any Zariski open subset $U\subset X$ the ring of sections ${𝒪}_X(U)$ of the structure sheaf over $U$ is an integral domain.

Basic fact: a scheme is integral iff it is both reduced and irreducible. Integral schemes of finite type over the spectrum of an algebraically closed field correspond (in the sense of equivalence of categories) to classical algebraic varieties.