What exactly does the weight filtration in Hodge theory have to do with the Weil conjectures? Let $X$ be a variety over $\mathbb{C}$, say separated. According to Deligne's results, there is a "mixed Hodge structure" on the total cohomology $H^\bullet(X(\mathbb{C}), \mathbb{Z})$. One component of this is a "weight filtration" on $H^\bullet(X(\mathbb{C}), \mathbb{Q})$. I haven't read Deligne's "Theorie de Hodge" and don't really understand all this, but I believe that in the case of a smooth projective variety, this reduces to usual Hodge theory and the weight filtration is the filtration by grading, and the extension to singular varieties comes by some sort of simplicial resolution by smooth objects.
Let $Y_0$ be a variety over a finite field $\kappa$. Given a mixed perverse sheaf $K_0$ on $Y_0$, there is a canonical (and functorial) weight filtration on $K_0$, such that the sucessive subquotients are pure complexes of increasing weight (in the sense of Weil II). 
What do these to have to do with each other? In section 6 of BBD (asterisque 100), it seems that the authors are using the functoriality of the weight filtration over finite fields to deduce results about the weight filtration over $\mathbb{C}$. Namely, I'd be interested if, given a perverse sheaf $K$ (say, of geometric origin) on a smooth, proper scheme $X$ over $\mathbb{C}$ which can be "spread out" to perverse sheaves of "reduction of $X$ mod a prime*" there is some way in which the weight filtration on the cohomology of $K$ (actually, I'm not sure that this exists, it seems to  in the constant case at least) can be 
viewed as a completion of the weight filtrations in finite characteristic.
Here is the specific result in BBD: Let $f: X \to Y$ be a separated morphism of schemes of finite type over $\mathbb{C}$. Suppose that the stalks of  $R^n f_* \mathbb{Q}$  are $H^n(X_y, \mathbb{Q})$, and that these form a local system.  Then the weight filtration on these stalks form a locally constant filtration of the local system $R^n f_* \mathbb{Q}$.
This appears to be proved by reducing mod a prime, where one has a Frobenius and the perverse weight filtration makes sense.
(One reason to think these might be related is that if $X_0$ is a proper smooth scheme over $\mathbb{\kappa}$, then the cohomologies $H^i(X, \mathbb{Q}_l)$ have weight $i$ by the Weil conjectures, and this has some correspondence with how the weight filtration was defined for projective, smooth schemes over $\mathbb{C}$.)
*Which is done by reducing the field $\mathbb{C}$ of definition to some finitely generated ring over $\mathbb{Z}$, and then working from there.
 A: see Deligne's ICM 1970 address (Theorie de Hodge I) as well as his ICM 1974 address.
A: Dear Akhil,
This is a big topic, although one that has been discussed at various times here, e.g.
In what setting does one usually define mixed sheaves and weights for them?
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:
http://www.math.purdue.edu/~dvb/preprints/tifr.pdf
