Dear Akhil,
This is a big topic, although one that has been discussed at various times here, e.g.
The idea is that for constant coeffients smooth projective varieties should be cohomologically the simplest. And by approximating more general varieties by these, via simplicial techniques etc., we get a weight filtration on cohomology which measures the deviation from the simplest case.
How to make this precise? Well
- In positive characteristic, we can say that smooth projective varieties are one on which Frobenius acts with expected bounds eigenvalues. So the weight filtration is defined via eigenspaces of these.
- Over $\mathbb{C}$, smooth projective varietes carry classical Hodge decompositions. The weight filtration needs to be (nontrivially) inserted into this picture via mixed Hodge theory.
The compatibility of the weights comes either by construction* or via the (somewhat conjectural) story of mixed motives.
For perverse coefficients, the story is already much more complex. The "simplest" cases should be intersection cohomology complexes with coefficients in direct images of families of smooth projective varieties. The analogue of (1) is BBD, and of (2) is Saito's theory that Uhlirch mentions.
*(Added) Perhaps I can say what I mean "compatible by construction". I'll take two examples, which give a sense of what's going behind the scenes.
A) take $X$ to be the complement of two points $p,q$ in smooth projective curve $\bar X$. Then have an exact sequence $$ 0\to W_1= H_1(\bar X, \mathbb{Q}) \to W_2=H^1(X, \mathbb{Q})\to \mathbb{Q}(-1)\to 0$$ The last map can be thought of as sort of residue at $p$. The symbol $\mathbb{Q}(-1)$ means the one dim vector space shifted into weight $2$, so this sequence also displays the weight filtration, There is an entirely analogous sequence in the $\ell$-adic world which gives the weights there. So these are compatible (pretty much by design).
B) For the second example, let us use $\bar X$ as above but with coefficients in the intersection cohomology $L=j_\ast R^i f_\ast\mathbb{Q}$, where $f:Y\to X$ is smooth projective. Then $L$ carries variation of Hodge structure of weight $i$. By Zucker [Ann. Math 1979] $H^1(\bar X, L)$ has a pure Hodge structure of weight $1+i$. In the $\ell$-adic world, the analgous statement is Deligne's purity theorem [Weil II]. Note that Zucker's theorem was one of the key analytic inputs in Saito's work, analogous to the role of Weil II in BBD.
Some References: Matt is correct that Saito's work isn't easy to get into. Aside from some expositions by Saito, I might suggest looking at the last few chapters of Peters and Streenbrink's book on mixed Hodge theory, which gives a pretty good introduction. I'm also linking my own, not quite successful, attempt to go through some of this:
