A Kähler manifold is a complex manifold for which the exterior derivative of the fundamental
form 
associated with the given Hermitian metric vanishes,
so 
. In other words, it is a complex
manifold with a Kähler structure. It
has a Kähler form, so it is also a symplectic
manifold. It has a Kähler metric, so it
is also a Riemannian manifold.
The simplest example of a Kähler manifold is a Riemann surface, which is a complex manifold of dimension 1. In this case, the imaginary part of any Hermitian metric must be a closed form since all 2-forms are closed on a two real dimensional manifold.
