For a given field $k$, let $A$ be an associative, unital $k$-algebra. Let $M$ be an $A$-bimodule. Define the Hochschild cochain complex of $A$ with coefficients in $M$ as
$$C^n(A;M):=\operatorname{Hom}_k(A^{\otimes n},M)$$ with the usual differential. One sometimes considers normalized $n$-cochains, namely those $f\in C^n(A;M)$ with $f(a_1\otimes \ldots \otimes a_n)=0$ if $a_i=1_A$ for some $i=1,\ldots, n$.
Why are these cochains called normalized?