A category of pure motives is informally a certain linearization or abelianization of a category of (say smooth projective) algebraic varieties over a field.
See also motive
THE FOLLOWING IS NOT A DEFINITION, just a rough description of steps which require lots of further details to become a definition.
Given an adequate equivalence relation for algebraic cycles (the typical 4 choices are the rational, numerical, algebraic and homological equivalences), one constructs a version of Chow rings, or what is axiomatically called a Weyl cohomology theory? for varieties; the treatment of adequate relations typically involves a nontrivial tool, the so called Chow moving lemma?. To define the category of effective pure motives, one starts with the category of smooth varieties over a field , then enlarges the hom-sets between any two objects by including a linear version of correspondences, which are roughly the Chow cohomology classes of cycles on . This gives an additive category; its Karoubian envelope with formally inverted Lefschetz motive? is the category of effective pure motives over . The Lefschetz motive is inverted in order to make the tensor category of motives rigid.
James S. Milne, Motives – Grothendieck’s Dream
Mihnyong Kim, Classical Motives: Motivic -functions
Bruno Kahn, pdf slides on pure motives