I will begin by giving a rough sketch of my understanding of motives.
In many expositions about motives (for example, www.jmilne.org/math/xnotes/MOT102.pdf), the category of motives is defined to be a category such that for every Weil cohomology (viewed as a functor) factors through it. This does not define the category uniquely, nor does it imply that it exists.
There are two concrete candidates that we can construct. The category of Chow motives, which is well-defined, is trivially a category of motives. However, it has some bad properties. For example, it is not Tannakian. The second candidate is the category of numerical motives. It too is well-defined, however it is only conjectured that it is category of motives (i.e., that every Weil cohomology factors through this category). This conjecture is closely related to (or perhaps even equivalent to?) Grothendieck's standard conjectures. That would be desirable, because the category of numerical motives is very well-behaved.
Furthermore, the original motivation for motives is that Grothendieck has proven that if the category of numerical motives is indeed a category of motives, then the Weil conjectures are correct.
So far, even though I a murky on many of the details, I follow the storyline.
Where does "motivic cohomology" (in the sense of, for example, www.claymath.org/library/monographs/cmim02.pdf) fit into this story?
I know that motivic cohomology has something to do with Milnor K-theory, but that is more or less where my understanding of the context of motivic cohomology ends. If motives are already an abstract object that generalizes cohomology, what does motivic cohomology signify? What is the motivation for defining it? What is the context in which it arose?