One of the most well known classes of K"ahler-Einstein manifolds, i.e. manifolds which carry a Kahler metric $g$ such that $Ric_{g}= \lambda \cdot g$ $c\in\mathbb{R}$, are the generalized flag manifolds $$G^{\mathbb{C}}/P\cong G/K$$ of a compact connected simple Lie group. Here $P$ is a parabolic subgroup of the complexification $G^{\mathbb{C}}$ of $G$, and $K=p\cap G$ is the centralizer of a torus $S\subset G$, i.e. $K=C(S)$. If $S=T=$maximal torus, then we obtain a full flag manifold $G/T$.
In the family of generalized flag manifolds, we find a very important subfamily of K"ahler-Einstein manifolds, the (isotropy irreducible) Hermitian symmetric spaces $M=G/K$ of compact type (i.e. compact symmetric spaces endowed with a Hermitian structure invariant under the symmetries. In particular, this Hermitian structire is Kahler) . It well know that such a space $M=G/K$ admits a unique (as isotropy irreducible) K"ahler-Einstein metric. We mention here two facts for isotropy irreducible Hermitian symmetric spaces $M=G/K$:
The isotropy subgroup $K$ has an 1-dimensional center.
They are the only generalized flag manifolds which are the same time symmetric spaces.
Now, a (generalized) flag manifold is also a homogenepous K"ahler manifold (the K"ahler structure on these manifolds arising form the fact that any flag manifold is an adjoint orbit of an element in the Lie algebra of $G$). In particular, flag manifolds exhaust all compact simply connected Homogeneous Kahler manifolds corresponding to (compact, connected) simple Lie groups. Their classification is based on the painted Dynkin diagrams.
Now, any $M=G^{\mathbb{C}}/P=G/C(S)$ admits a finite number of invariant complex structures. More-ever for any such complex structure we can define (a unique) homogeneous K"ahler--Einstein metric, which given in terms of representation theory by computing the Koszul form $2\delta_{\frak{m}}$. Thus a flag manifolds admits a finite number of K"ahler-Einstein metrics, Note that if some of the invariant complex structures are equivalent, then, the K"ahler-Einstein metrics corresponding to these complex structures would be isometric.
More information about the geometry of flag manifolds, painted Dynkin diagrams, K"ahler-Einstein metrics, etc, can be found in the following article
D. V. Alekseevsky: Flag manifolds, in Sbornik Radova, 11th Jugoslav. Geom. Seminar. Beograd 6 (14) (1997) 3--35.
D. V. Alekseevsky and A. M. Perelomov: Invariant K"ahler-Einstein metrics on compact homogeneous spaces, Funct. Anal. Appl. 20 (3) (1986) 171--182.