I was reading HTT 2.1.4, and I just totally lost what was going on. Could someone provide some motivation for this section? Why do we want another model structure?
I'm sorry for not providing motivation, but as you can see, I'm unable to do so.
Here's a link to the ArXiv version: http://arxiv.org/pdf/math/0608040v4
If you have a category $C$, then you can consider the category of functors $Func(C,Sets)$ from $C$ to sets. This comes with the Yoneda functor, $C^{op}\to Func(C,Sets)$, and is a generally useful thing to think about.
In $(\infty,1)$-category land, you want to start with an $(\infty,1)$-category $S$, and build the $(\infty,1)$-category of functors $Func(S,Spaces)$ from $S$ to spaces.
Lurie is describing one way to do this. Given a quasicategory $S$, he produces a closed model category $(sSet/S, covariant)$ on the slice category $sSet/S$ which "models" this functor category.
The analogy is to the "point category" construction. Given a functor $F:C\to Sets$, you can build a category $P_F$, whose objects are pairs $(c,x)$ where $c$ is an object of $C$, and $x\in F(c)$, and maps $(c,x)\to (c',x')$ are $f:c\to c'$ in $C$ such that $Ffx=x'$. Such a thing comes with a forgetful functor $U:P_F\to C$, and one can produce an equivalence of categories $$ Func(C,Sets) \qquad \Leftrightarrow\qquad (\text{certain full subcategory of $Cat/C$}). $$ The subcategory is that of functors $U:D\to C$ such that given $f:c\to c'$ in $C$ and $d\in D$ such that $U(d)=c$, there is a unique $g: d\to d'$ in $D$ such that $U(g)=f$.
So, fibrant objects in Lurie's covariant model category for $sSet/S$ are supposed to look like left Grothendieck fibrations, and this model category should model "$Func(S,Spaces)$".
Edit: originally, I said that the certain full subcategory was that of "left Grothendieck fibrations", but I don't think that's right. You can play the above story a bit more generally, with $Sets$ replaced by $Groupoids$; the functor $P_F\to C$ corresponding to a (pseudo)functor $F:C\to Groupoids$ is a typical example of what's called a "left Grothendieck fibration" (or "fibered groupoid"), consult nLab for details.
The quick answer is the covariant model structure on sSet/S is one way to build an infinity-category of infinity-copresheaves on S, when S is an infinity-category. In fact, I don't understand why the covariant model structure is introduced first rather than the contravariant one- it is the later which is used to construct infinity-presheaves, which are of course important examples of infinity-topoi.
(I am using the terminology "infinity-presheaf" to mean a contravariant infinity-functor from S to the infinity-category of infinity-groupoids)
It would perhaps be best to recall what happens in the case of 2-categories:
Let C be a category. We can, on one hand, consider the bicategory of weak functors C^op->Gpd, where the target is the bicategory of groupoids. On the other hand, we can consider categories fibred in groupoids over C, that is a functor D->C which is a Grothendieck fibration in groupoids. Both of these objects, weak presheaves and fibred categories respectively, naturally form bicategories.
We have a 2-functor G:Gpd^{C^op}->Fib_Gpd(C) between these bicategories given by the "Grothendieck construction". It has a left 2-adjoint and together this adjoint pair forms an equivalence of bicategories.
Lurie proves the infinity-analogue of this statement. To do so, he needs to form an infinity category of "Grothendieck fibrations in groupoids", and an infinity category of "infinity presheaves" and show they are equivalent.
The infintiy-version of Grothendieck fibration in groupoids is what Lurie calls a "right fibration". In particular, C->D is a Grothendieck fibration in groupoids if and only if N(C)->N(D) is a right fibration. Also, the fibers of any right-fibration are Kan-complexes, hence, infinity groupoids.
Given a simplicial set S, the contravariant model structure on sSet/S is enriched in sSet_Quillen so we can form the associated full simplicial category on fibrant and cofibrant objects. An an object X->S is fibrant in this model structure if and only if it is a right-fibration. Hence, the homotopy-coherent nerve of this simplicial category is the infinity-category of "Grothendieck fibrations in infinity groupoids over S".
In 2.2.1, Lurie introduces a functor St:sSet/S->sSet^{C(S)^op} where C is the left-adjoint to the homotopy-coherent nerve- here we mean we are considering functors of simplicial categories (treating sSet as a simplicial category since it is enriched over itself). Since we can identify sSet/S with Set^{(\Detla/S)^op} and the functor is colimit preserving, by formal nonsense it has a right adjoint which Lurie denotes by "Un". "Un" is the "infinity-Grothendieck construction".
We can now equip sSet^{C(S)^op} with the projective model structure, and then the adjoint pair (St,Un) forms a Quillen-equivalence.
Now, sSet^{C(S)^op} can be turned into an infinity-category by applying the same construction I said before- treat it as a simplicial category and restrict to fibrant and cofibrant objects, and take the homotopy-coherent nerve. This infinity-category is the infinity-category of infinity-presheaves on S.
The Quillen-equivalence (St,Un) turns into an adjunction between the infinity-category of right-fibrations over S, and the infinity-category of infinity-presheaves over S, and moreover is an equivalence of infinity-categories.
The upshot is, the contravariant model structure gives us another way of describing infinity-presheaves.
As a side note, to understand why the contravariant model structure is defined the way it is, you should look at how the functor St is defined- the model structure is essentially "designed" to so that (St,Un) becomes a Quillen equivalence.
