A Hodge structure can be defined as a real, algebraic representation of the Deligne torus ${Res}^\mathbb{C}\_{\mathbb{R}}\mathbb{G}\_m$. Coming from Kahler manifolds the intuition for this is clear. The complex structure on the smooth tangent bundle of a Kahler manifold is itself a sort of parametrized hodge structure (of weight -1) and so are the exterior powers of the cotangent bundle. 


For the definition of a polarized Hodge structures however I can find almost no intuition. All the definitions I saw so far seem to suggest that a polarization is a sort of bilinear form inside the category of hodge structures with some "extra conditions". I won't pronounce the definition since I can't say I have a specific problem with anything, I only wish there was a way to motivate it. 


For instance: 


1. Is the polarization related to the intersection form coming from ordinary cohomology? If so how?
2. What is polarization good for? Why isn't the intersection form enough? Does the polarization carry more information? If so what kind?
3. Is there a canonical way to define it similar to the definition of hodge structure I gave above? Ideally such a way would make all the sign issues in this definition transparent, and would be invariant to the convention one uses for the tate twist $\mathbb{R}(n)$. (whether you twist by $2\pi i$ or not).
4. Do polarization come up in $l$-adic setting? If so how?



Intuition for polarized Hodge structures
Perhaps it's not obvious at first, but the notion of polarization is important. Some examples, where it comes up:


1. A complex torus $X=\mathbb{C}^n/L$, where $L$ is a lattice, is compact Kähler, so $H^1(X,\mathbb{Z})$ carries a Hodge structure. A classical theorem of Riemann can be understood as saying that $X$ is an abelian variety (i.e. $X$ "comes from" algebraic geometry) if and only if this Hodge structure carries a polarization.
2. If $L$ is a local system underlying a *polarizable* variation of rational Hodge structure, whatever that means, over a quasiprojective base then the monodromy representation is semisimple. In this generality, the result is due to Schmid and polarizations are essential.


