The cellular approximation theorem states that every continuous map between CW complexes (with chosen CW presentations) is homotopic to a cellular map? (a map induced by a map of cell complexes).
This is a CW analogue of the simplicial approximation theorem? (sometimes also called lemma): that every continuous map between the geometric realizations of simplicial sets is homotopic to a map induced by a map of simplicial sets.
wikipedia: cellular approximation
E. H. Spanier, Algebraic topology, Springer 1966, ch. 7. sec. 6
A. Hatcher, Algebraic topology, Cambridge Univ. Press 2002 , link