A very nice and very general version of Tannakian formalism is in Jacob Lurie's paper, Tannaka duality for geometric stacks, arXiv:math/0412266.
I like to think of Tannaka duality as recovering a scheme or stack from its category of coherent (or quasicoherent) sheaves, considered as a tensor category. From this POV the intuition is quite clear: having a faithful fiber functor to Vect (or more generally to R-modules) means your stack is covered (in the flat sense) by a point (or by Spec R). This is why you get (if you're over an alg closed field) that having a faithful fiber functor to Vect_k means you're sheaves on the quotient BG of a point by some group G, i.e. Rep G.. Over a more general base, you only locally look like a quotient of Spec R (or Spec k for k non-alg closed) by a group ---- ie you're a BG-bundle over Spec R, aka a G-gerbe. Even more generally, the kind of Tannakian theorem Jacob explains basically says that any stack with affine diagonal can be recovered from its tensor category of quasicoherent sheaves..
Actually the construction of the stack from the tensor category is just a version of the usual functor Spec from rings to schemes. Recall that as a functor, Spec R (k) = homomorphisms from R to k. So given a tensor category C let's define Spec C as the stack with functor of points Spec C(k) = tensor functors from C to k-modules (for any ring k, or algebra over the ground field etc). The Tannakian theorems then say for X reasonable (ie a quasicompact stack with affine diagonal), we have X= Spec Quasicoh(X) --- so X is "affine in a quasicoherent-sheaf sense". Again, the usual Tannakian story is the case X=BG or more generally a G-gerbe.