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?

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Second-order approximation to a potential function at a critical point? –  Qiaochu Yuan Mar 18 '11 at 20:27
I think the question is a little broad. Can you be more specific about what you are looking to gain by having such a list? –  Qiaochu Yuan Mar 18 '11 at 20:34
People are quite fond of :  Symmetric bilinear forms [by] J. Milnor [and] D. Husemoller by John Willard Milnor, 1973,Springer-Verlag edition, in English. –  Will Jagy Mar 18 '11 at 21:10
What do you mean by "arise naturally"? In your examples you are often introducing constraints on a dimension to force a 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 usually not a quadratic form. (Continued...) –  KConrad Mar 18 '11 at 22:11
I would consider as examples that give incentive to study general quadratic forms those constructions which, in general, produce quadratic forms in potentially any number of variables, not just a small number because you fixed some parameter to force a quadratic form. For instance, the trace form on an associative algebra or the Killing form on a Lie algebra. These are symmetric bilinear forms in possibly many variables (depends on the dimension of the algebra) and correspond to some quadratic forms in many variables. –  KConrad Mar 18 '11 at 22:14

Dear Olivier, in line with the more advanced nature of this site, let me give an example of a less elementary nature.

Consider a compact Riemann surface $X$ of genus 2 and on it stable vector bundles $E$ of rank 2 whose determinant bundle $\Lambda ^2E$ is isomorphic to some fixed line bundle $L$ of degree $-1$. Newstead has proved that the moduli space of those vector bundles is the intersection of two quadrics in five-dimesional projective space $\mathbb P^5(\mathbb C)$. And one of those quadrics is the Klein quadric in $\mathbb P^5(\mathbb C)$ parametrizing the lines in some three-dimensional projective space canonically associated to $X$ and $L$. (A Klein quadric is the quadric you mention in number (v) of your list.)

References
P E. Newstead Stable bundles of rank 2 and odd degree over a curve of genus 2, Topology 7 (1968), 205-215.
For a geometric description including the role of the Klein quadric, see:
M. S. Narasimhan and S. Ramanan Moduli of Vector Bundles on a Compact Riemann Surface, Annals of Mathematics, Vol. 89, No. 1, 1969 , pp. 14-51.

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@Olivier: when you ask such a broad, open-ended question, it seems to be good form to wait more than a couple of hours to accept an answer. –  Pete L. Clark Mar 19 '11 at 5:55
you're right, what I meant to do is to give his answer a thumbs up, not close the question. –  Olivier Bégassat Mar 20 '11 at 0:03
can you explain please? I looked up "fixed points of Moebius transforms" and only found that (the non identity ones) they either fix two points on the boundary $\partial\mathbb{H}$, one point on the boundary and no other, or two points in $\mathbb{H}$ –  Olivier Bégassat Mar 20 '11 at 0:28