The study of homogeneous varieties is a vast part of algebraic geometry, notably of algebraic group theory.
You certainly know quite a few homogeneous projective varieties. As you said there are abelian varieties, which are groups, but also quotients of groups like projective spaces, grassmannians and flag varieties.
The quotients here are quotients of affine algebraic groups by subgroups which are not normal, hence the quotient varieties are not groups. On the other hand, since you want the quotient to be projective, you have to divide out by a parabolic subgroup.
The definition of parabolic is a bit technical, but to give you some feeling for the notion, let me mention that given a finite dimensional vector space $V$, a parabolic subgroup of $GL(V)$ is a subgroup fixing some (incomplete) flag $V_0 \subsetneq V_1\subsetneq ...\subsetneq V_s\subsetneq ...\subsetneq V_r=V \quad $ [incomplete means that you don't require $dimV_s=s$]
As to dimension two, Tits ( the celebrated Belgian Abel prize laureate) has classified not only the projective homogeneous surfaces but also the compact holomorphic ones here ( Potters has generalized this classification to almost homogeneous compact holomorphic surfaces there)