Let $C$ be a dg-coalgebra, $A$ a dg-algebra, $\tau:C\to A$ the twisting cochain, $L$ a right $C$-dg-comodule with coaction$\delta_L:L \to L\otimes C$ and $M$ a left $A$-dg-module with action $m_M:M\otimes A\to A$. The twisted tensor product$L\otimes_\tau M$ is the chain complex that coincides with the ordinary tensor product$L\otimes M$ as a graded module over the ground ring, and whose differential $d_\tau$ is given by

Edgar H. Brown Jr. Twisted tensor products I, Annals of Math. (2) 69 (1959) 223–246 doi

V. A. Smirnov, Simplicial and operadic methods in algebraic topology, Translations of mathematical monographs 198, AMS, Providence, Rhode Island 2001.

Kenji Lefèvre-Hasegawa?, Sur les $A_\infty$-catégories, thesis, (Université Denis Diderot – Paris 7, Paris, November 2003). Corrections, by B. Keller, available here.