An projective algebraic variety (over an algebraically closed field k) is the 0-locus of a homogeneous? ideal of polynomials in (n+1) variables over k in the projective n-space ℙ n.
The archetypical example is projective space itself. In direct generalization of this but less evident: every Grassmannian is naturally a projective variety, even a smooth variety.
This generalizes to quiver representations. Every Grassmannian of a Quiver representation is a projective variety. In fact, every projective variety arises this way (Reineke, ‘12).