I am always fascinated when a quadratic form (or a quadric) arises naturally. I have some elementary examples, but most of all, I want to learn more examples. I hope this question isn't considered too vague for MO. Most forms I list are really elementary, and all are finite dimensional.

I got most of the following examples from M.Berger, Geometry I & II, and from the truly beautiful book "Eléments de géométrie : actions de groupes" by french author Rached Meinmné.

$(0)$ the discriminant on the affine space of unitary degre 2 polynomials

$(i)$ the determinant on endomorphisms of a 2 dimensional vector space, and $\mathrm{Tr}^2-4\mathrm{det}$

$(ii)$ the radical on the space of quadratic forms on a 2 dimensional vector space, and the isotrope cone (not sure about the name, degenerate cone?).

$(iii)$ the family of hermitian forms (built from the Wronskian) on the solution space of the discrete Schroedinger equation that allow one to show the existence of right and left side $L^2$ solutions, and the Weyl m function.

$(iv)$ If $\Delta$ is any $2$ dimensional complex vector space, then $\mathrm{Herm}(\Delta)$, the real vector space of hermitian forms on $\Delta$, carries a natural quadratic form obtained by constructing an essentially unique morphism $\rho$ from $\mathrm{Herm}(\Delta)$ to $\mathrm{Hom}(\Delta\oplus\overline{\Delta})$ such that for all $h\in\mathrm{Herm}(\Delta),~\rho(h)^2$ is proportional to $\mathrm{Id}$, the proportionality defining the quadratic form. Here, $\rho$ only depends on a choice of a nonzero element $\omega\in\Lambda^2\Delta^*$.

$(v)$ If $V$ is a 4 dimensional vector space, then $\Lambda^2 V$ carries the natural quadric $Q(v)=v\wedge v$ where $\Lambda^4 V$ is identified with the underlying field, which vanishes exactly when $v$ comes from the canonical map $\mathrm{Gr}(2,V)\rightarrow P\Lambda^2V$.

I remember reading about one on the space of circles, but I forgot the details. What other examples of natural quadratic forms are there?

forcea quadratic form in what otherwise would really be better considered as a homogeneous polynomial of some degree. For instance, the determinant on endomorphisms of an n-dim. space is a homogeneous polynomial of degree n in n variables. To say the determinant becomes a quadratic form when you set n = 2 seems, to me, to be missing the big picture of what happens in general, since the determinant is usuallynota quadratic form. (Continued...) – KConrad Mar 18 '11 at 22:11