Compact complex manifolds are holomorphically convex.Stein's original definition had in addition to holomorphic convexity separation of points by holomorphic functions and local coordinates given by holomorphic functions  .Stein himself called these holomorphically complete manifolds  .He introduced it to study the solution of Cousin problems on complex manifolds  . Remmert showed that a noncompact connected complex manifold is holomorphically convex iff it admits a proper holomorphic mapping to a Stein space  . You can find some of these discussed in the book of Grauert and Remmert Theory of Stein spaces  . From Stein's definition one can see that Stein manifolds admit strictly plurisubharmonic exhaustion functions  .The converse is a deep theorem of Grauert.

Compact complex manifolds are holomorphically convex. Stein's original definition had in addition to holomorphic convexity separation of points by holomorphic functions and local coordinates given by holomorphic functions. Stein himself called these holomorphically complete manifolds. He introduced it to study the solution of Cousin problems on complex manifolds. Remmert showed that a noncompact connected complex manifold is holomorphically convex iff it admits a proper holomorphic mapping to a Stein space. You can find some of these discussed in the book of Grauert and Remmert Theory of Stein spaces.

From Stein's definition one can see that Stein manifolds admit strictly plurisubharmonic exhaustion functions. The converse is a deep theorem of Grauert.