Why would the category of Motives be Tannakian? After reading the answer to my previous question: What are the different theories that the motivic fundamental group attempts to unify?
I decided to read up on Tannakian formalism.
Given the category of numerical motives, and assuming Conjecture C of the standard conjectures (the one regarding the grading of numerical motives), one can construct a category that will be Tannakian. This will be done by changing the sign of the ``canonical'' morphism $h^iX\otimes h^jX \cong h^jX \otimes h^iX$ for $ij$ odd .
It seems in texts about motives, that the end goal was always to achieve a Tannakian category. But what motivation is there for this? Why would a category that has to do with motives be the category of representations of an affine group scheme? This seems crazy to me. Is this immitative of some easier, more well-understood, theory in which it make sense to relate cohomology with representations?
Also, is it conjectured what this mysterious affine group scheme is, in the case of numerical motives with the adjustment written above?
 A: Actually, the category of motives isn't equivalent to the category of representations of an affine group scheme except in characteristic zero, and even there the equivalence depends on the choice of a fibre functor.
The functor to motives is supposed to be a universal cohomology theory. Certainly, one would like the target of a cohomology theory to be at least tannakian.
If you assume the Hodge conjecture, then the affine group scheme attached to the category of abelian motives over $\mathbb{C}$ (that generated by abelian varieties) is more-or-less known --- at least its algebraic quotients are classified.
If you assume the Tate conjecture, then the affine groupoid attached to the category of motives over an algebraic closure of $\mathbb{F}_p$ is more-or-less known.
In the general case nothing is known except that the group is VERY BIG --- for example, over $\mathbb{C}$ it has uncountably many distinct quotients isomorphic to PGL(2).
A: My impression is that the language of Tannakian categories is a nice thing for itself; thus applying it to motives may (and will, see below) yield interesting applications. There are several ways of dealing with Tannakian categories, and I am not sure that dealing with explicit affine group schemes appearing this way is the "main" one (since this group scheme is awfully huge and complicated for motives, and you have to "make the category neutral" to ensure its existence). Also, Tannakian categories give a possibility of dealing with the category of motives "abstractly".
Probably the nicest results on the relation of motives to Tannakian formalism are in the case where the base field is either finite or (at least) algebraic over a finite field; you can find them in Milne's http://www.jmilne.org/math/articles/1994aP.pdf. Firstly, the category of numerical motives is known to be Tannakian in this case; see Proposition 1.1. To say more about it one needs certain "standard" conjectures. In particular, the Tate conjecture allows to describe the category of motives over a finite field almost completely in Corollary 1.16, Proposition 1.17 (these two statements are improved in Propositions 3.7 and 3.8), Proposition 2.6, and Proposition 2.22. Moroever, Theorem 3.13 (cf. also Theorem 3.19) gives a complete description of motives over the algebraic closure $\mathbb{F}$  of a finite field. Lastly, Theorem 4.22 gives a very funny functor from the so-called CM-motives over the algebraic closure of $\mathbb{Q}$ into motives over $\mathbb{F}$; this result crucially depends on the language of Tannakian categories (since no "geometric" description of this functor is given).
