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A string vertex operator inserts an asymptotic string state at a point of the worldsheet. Its conformal weights encode the physical-state and mass-shell conditions.

These insertions are described by the state-field correspondence of a vertex operator algebra; an individual insertion is not the whole algebra.

A vertex operator algebra is an integer-graded vector space equipped with a state-field correspondence $a\mapsto Y(a,z)$, a vacuum, translation operator and conformal vector satisfying vacuum, translation, locality and compatibility axioms. It packages the local operator product expansions of a chiral conformal field theory. A string vertex operator is an individual state insertion, rather than this complete algebraic structure.