We introduce notions of open-string vertex algebra, conformal open-string
vertex algebra and variants of these notions. These are
``open-string-theoretic,'' ``noncommutative'' generalizations of the notions of
vertex algebra and of conformal vertex algebra. Given an open-string vertex
algebra, we show that there exists a vertex algebra, which we call the
``meromorphic center,'' inside the original algebra such that the original
algebra yields a module and also an intertwining operator for the meromorphic
center. This result gives us a general method for constructing open-string
vertex algebras. Besides obvious examples obtained from associative algebras
and vertex (super)algebras, we give a nontrivial example constructed from the
minimal model of central charge c=1/2. We establish an equivalence between the
associative algebras in the braided tensor category of modules for a suitable
vertex operator algebra and the grading-restricted conformal open-string vertex
algebras containing a vertex operator algebra isomorphic to the given vertex
operator algebra. We also give a geometric and operadic formulation of the
notion of grading-restricted conformal open-string vertex algebra, we prove two
isomorphism theorems, and in particular, we show that such an algebra gives a
projective algebra over what we call the ``Swiss-cheese partial operad.''