Complex algebraic varieties have this property, but the converse is not true: Hironaka's example gives a smooth 3-dimensional Moishezon manifold that is not an algebraic variety or scheme. Moishezon (1967, Chapter I, Theorem 11) showed that a Moishezon manifold is a projective algebraic variety if and only if it admits a Kähler metric. Artin (1970) showed that any Moishezon manifold carries an algebraic space structure; more precisely, the category of Moishezon spaces (similar to Moishezon manifolds, but are allowed to have singularities) is equivalent with the category of algebraic spaces that are proper over Spec(C).
Moishezon, B.G. (1967). "On n-dimensional compact varieties with n algebraically independent meromorphic functions, I, II and III (1966) (English translation version)". Seven Papers on Algebra, Algebraic Geometry and Algebraic Topology. American Mathematical Society Translations: Series 2. Vol. 63. doi:10.1090/trans2/063. ISBN9780821844335.