For example on wikipedia I found that one can also define a Hermitian manifold as a real manifold with a Riemannian metric that preserves a complex structure.
For real manifold it should be understood a manifold with real charts, i.e., diffeomorphisms from an open subsets of M and open subsets of $\mathbb{R}^n$, then what does it mean with complex structure? is only the complexification of its tangent space?
Best Answer
In this sense, a complex structure is an endomorphism of the tangent bundle $J:TM\to TM$ such that $J^2=-1$ (it plays the role of the multiplication by $\sqrt{-1}$). Saying that a Riemannian metric $g$ preserves the complex structure amounts to saying that $g(Jv,Jv)=g(v,v)$ for all tangent vector $v$.
Note that this forces the (real) dimension of $M$ to be even, and that if $M$ is a complex manifold, taking holomorphic charts $(U,\varphi=(z^1,\dots,z^n)=(x^1,y^1,\dots,x^n,z^n))$ and defining locally a $J_0:TM\to TM$ by $J_0(\frac{\partial}{\partial x^i})=\frac{\partial}{\partial y^i}$ and $J_0(\frac{\partial}{\partial y^i})=-\frac{\partial}{\partial x^i}$ defines a canonical complex structure.
Moreover, given any $J$, the complexification of the (real) tangent bundle of $M$ is written as $TM\otimes\mathbb{C}:=T_\mathbb{C}M=T^{(1,0)}M\oplus T^{(0,1)}M$ where $T^{(1,0)}M$ is the vector bundle associated to the eigenvalue $i$ of $J_\mathbb{C}$ (where $J_\mathbb{C}(v\otimes z)=J(v)\otimes z$) and $T^{(0,1)}M$ is the vector bundle associated to the eigenvalue $-i$ of $J$. Thus you have a $\mathbb{C}$-linear isomorphism between $(TM,J)$ and $T^{(1,0)}M$ as complex bundles (and a $\mathbb{C}$-antilinear one between $(TM,J)$ and $T^{(0,1)}M$).