The functor of points as introduced and defined by Grothendieck is either the image of some geometric space by Yoneda embedding into the category of presheaves on some ambient category of geometric spaces or, better, its restriction to some subcategory of local models or nice spaces. In another way, the functor of points corresponding to a space is its corresponding representable presheaf, but the point in the notion is that nonrepresentable functors of points are of the main interest, as well as their relation to representables (for example, being prorepresentable).
In this approach the spaces are sheaves of sets in some subcanonical Grothendieck topology on the category of local models . Not only spaces, but also additional structures on spaces (like group structure, equivariance, tangent bundle) are represented as presheaves of sets, of groups, of -modules etc. on .