Any Weil cohomology theory factors uniquely through the category of Chow motives, but the category of Chow motives itself is not a Weil cohomology theory, since it is not an abelian category.
A Weil cohomology theory is a contravariant functor satisfying the axioms below.
For each smooth projective algebraic variety X of dimension n over k, then the graded K-algebra is required to satisfy the following: There are four so-called classical Weil cohomology theories: The proofs of the axioms for Betti cohomology and de Rham cohomology are comparatively easy and classical.
The vanishing of Betti cohomology groups exceeding twice the dimension is clear from the fact that a (complex) manifold of complex dimension n has real dimension 2n, so these higher cohomology groups vanish (for example by comparing them to simplicial (co)homology).
The de Rham cycle map also has a down-to-earth explanation: Given a subvariety Y of complex codimension r in a complete variety X of complex dimension n, the real dimension of Y is 2n−2r, so one can integrate any differential (2n−2r)-form along Y to produce a complex number.