Weil introduced the term "polarization" in connection with his study of abelian varieties with complex multiplication. His definition is slightly different from what one sees today; one might call it a polarization up to isogeny instead of a polarization. One can find a discussion in Weil's article "On the theory of complex multiplication" ([1955d] in volume 2 of his Oeuvres Scientifiques). There are a few notes on the context of that article (and its relation to the work of Shimura and Taniyama) at the end of the volume. Weil says in the article itself (and again in the notes) that "the word 'polarization' is chosen so as to suggest an analogy with the concept of 'oriented manifold' in topology."
As Weil mentions in the article, Matsusaka (who might be described as a student of Weil) also did some important early work on polarized varieties apparently around the time that Weil invited him to Chicago (1954). Kollár, a student of Matsusaka, wrote a nice memorial article for the Notices in 2006, which discusses the work of Matsusaka, especially in relation to the theory of moduli. One can find Matsusaka's early work on polarizations in his "Polarized varieties, fields of moduli, and generalized Kummer varieties of polarized abelian varieties" in American J. of Math. vol. 80 No. 1, 1958. The introduction starts as follows:
"In studying theta-functions and abelian functions, we sometimes fix the set of scalar multiples of a principal matrix attached to the given Riemann Matrix. This implies that we fix one divisor class and scalar multiples of it with respect to homology on the corresponding complex torus. In this paper we shall introduce the corresponding notion in the abstract case, not only to abelian varieties, but also to arbitrary varieties as done in Weil [22]."
(Reference [22] is Weil's paper on complex multiplication mentioned above.)
For independent confirmation of Weil and Matsusaka as the origin of the notion of polarization, one may see various articles of Shimura from around the same time, where he repeatedly cites these two. For example, at the end of "Modules des variétés abéliennes polarisées et fonctions modulaires," he writes, "Les notions d'une variété polarisée, d'un corps du module et d'une variété de Kummer sont dues à Weil et Matsusaka; ce dernier a traité le cas général de variétés quelconques."
For intuition about polarizations, it may be helpful to think like this: the category of abelian varieties over a field $k$ (say up to isogeny) acts like a full subcategory of the category of representations of a group $G$ on $\mathbb{Q}$-vector spaces. (To show the shift in thinking we can write $V(A)$ when we think of $A$ as like a vector space.) The category of abelian varieties is too small to admit anything like the tensor products that exist in the larger category of $G$-representations, but one can see some multilinear structures. In particular, the dual abelian variety is like a dual representation---or at least a dual representation twisted by a character. A divisorial correspondence between abelian varieties $A$ and $B$ (a sufficiently rigidified line bundle on $A\times_k B$---a biextension of $(A,B)$ by $\mathbf{G}_m$) is like a $G$-invariant bilinear form on $V(A)\times V(B)$. A polarization on $A$ is a divisorial correspondence on $A\times_k A$ satisfying a symmetry and positivity condition (ampleness of pullback along the diagonal), which is like a $G$-invariant bilinear form on $V(A)\times V(A)$ satisfying a symmetry and positivity condition. There are some subtleties in the translation; for example, the symmetry on side of $A$ corresponds to antisymmetry on the side of $V(A)$. We know from the early pages of books on representation theory that $G$-invariant bilinear forms satisfying positivity conditions are useful---they give us complete reducibility, for example---and Weil used polarizations in a similar way in [1955d]. The fact that abelian varieties are polarizable corresponds to something like the fact that $G$ is a compact group, but the aforementioned subtleties mean that statement is not quite right.
These vague remarks can be made precise when $k = \mathbb{C}$ in the way that Francesco Polizzi suggests: take $V(A)$ to be $H_1(A^{\rm an},\mathbb{Q})$ equipped with its standard Hodge structure, and the above remarks correspond to some of the theory of theta functions (which is what Matsusaka had in mind in his introduction).
A final remark: I've always been a bit suspicious of how Weil justifies the polarization terminology, since there is a notion in classical projective geometry with a similar flavor called a "polarity." I wonder if Weil also had this classical terminology in mind when coining the term "polarization." (A polarity on the projective space associated to a vector space $V/k$ is a geometric structure related to a non-degenerate symmetric bilinear form $B:V\times V\to k$. A polarization on an abelian variety $A/\mathbb{C}$ is a geometric structure related to a certain type of alternating bilinear form on the period lattice of $A$. Both geometric structures are symmetric divisorial correspondences.)
Best Answer
It's a result related in spirit to Minkowski's theorem that $\mathbb Q$ admits no non-trivial unramified extensions. If $A$ is an abelian variety over $\mathbb Q$ with everywhere good reduction, then for any integer $n$ the $n$-torsion scheme $A[n]$ is a finite flat group scheme over $\mathbb Z$. Although this group scheme will be ramified at primes $p$ dividing $n$, Fontaine's theory shows that the ramification is of a rather mild type: so mild, that a non-trivial such family of $A[n]$ can't exist.
In the last 25 years, there has been much research on related questions, including by Brumer--Kramer, Schoof, and F. Calegari, among others. (One particularly interesting recent variation is a joint paper of F. Calegari and Dunfield in which they use related ideas to construct a tower of closed hyperbolic 3-manifolds that are rational homology spheres, but whose injectivity radii grow without bound.)
EDIT: I should add that the case of elliptic curves is older, due to Tate I believe, and uses a different argument: he considers the equation computing the discriminant of a cubic polynomial $f(x)$ (corresponding to the elliptic curve $y^2 = f(x)$) and shows that this solution equation has no integral solutions giving a discriminant of $\pm 1$.
This direction of argument generalizes in different ways, but is related to a result of Shafarevic (I think) proving that there are only finitely many elliptic curves with good reduction outside a finite set of primes. (A result which was generalized by Faltings to abelian varieties as part of his proof of Mordell's conjecture.)
Finally, one could add that in Faltings's argument, he also relied crucially on ramification results for $p$-divisible groups, due also to Tate, I think, results which Fontaine's theory generalizes. So one sees that the study of ramification of finite flat groups schemes and $p$-divisible groups (and more generally Fontaine's $p$-adic Hodge theory) plays a crucial role in these sorts of Diophantine questions. A colleague describes it as the ``black magic'' that makes all Diophantine arguments (including Wiles' proof of FLT as well) work.
P.S. It might be useful to give a toy illustrative example of how finite flat group schemes give rise to mildly ramified extensions: consider all the quadratic extensions of $\mathbb Q$ ramified only at $2$: they are ${\mathbb Q}(\sqrt{-1}),$ ${\mathbb Q}(\sqrt{2})$, and ${\mathbb Q}(\sqrt{-2})$, with discriminants $-4$, $8$, and $-8$ respectively. Thus ${\mathbb Q}(\sqrt{-1})$ is the least ramified, and not coincidentally, it is the splitting field of the finite flat group scheme $\mu_4$ of 4th roots of unity.