See the last section of Levine’s K-theory handbook chapter. He discusses “realization functors” for various kinds of categories of motives, and various cohomology theories. Seems really interesting.
Realizations can also refer to maps relating A1-homotopy theory to classical homotopy theory. See end of chapter 3 in Morel-Voevodsky: A1-homotopy theory of schemes.
K-th 992 : Réalisations des complexes motiviques de Voevodsky by Florence Lecomte and Nathalie Wach
In this paper, we construct realizations of motivic complexes over a number field. The De Rham realization is represented by a motivic De Rham complex and has a Hodge filtration. The Betti and l-adic realizations are integrally defined. When restricted to geometrical motives, the realization functors are endowed with Bondarko’s weight filtration and rationally agree with Huber’s realizations.
nLab page on Realizations