Not a fact but a philosophy: to me the most important way of thinking in algebraic geometry that would be useful in many areas is that of a moduli space. I.e. the idea that the set of isomorphism classes of certain objects should be viewed with structure of that same type, and its properties studied as a tool for understanding the original types of objects. This I believe is basic to the work of Chris Byrnes alluded to above. This philosophy is perhaps not due to or original with algebraic geometry, but is practiced systematically there. It may derive from algebraic topology, (classification of vector bundles, E.H. Brown's representability of cohomology,....), like many other things in AG.
It might be of interest e.g. to some high school students to know that Euclid proved the set of congruence classes of circles is an open half line, and that the set of all triangles modulo similarity is parametrized by the interior of an isosceles right triangle, modulo the reflection in the altitude on the hypotenuse (via the unordered coordinates AA given by the two largest angles), hence also the interior of an isosceles right triangle, but with the interior of one edge added in. Then the set of congruence classes of triangles can be seen as the product of this triangular region with an infinite open half line, i.e. an infinite parallelpiped, (via the ASA theorem). They might then compare this with the realization of this same space by the coordinates SSS.