In page 161 of Fulton's "Intersection Theory" is written:
Let $D\subseteq W \subseteq V$ be closed imbeddings of schemes. Assume that $D$ is a Cartier divisor on $V$. There is a closed subscheme $R$ of $W$, called the residual scheme of $D$ in $W$ (with respect to $V$), such that $W=D\cup R$ and, moreover, the ideal sheaves on $V$ are related by
$F(W)=F(D).F(R)\qquad(1)$
Indeed, the inclusion $D\subseteq W$ means that $F(W)\subseteq F(D)$, so that every local equation for $W$ is uniquely divisible by a local equation for $D$; the quotients give local equations for $R$.
I can't understand this definition. In particular the condition on ideal sheaves. How relation (1) tell us we can divide local equations? It is about product of ideals not of elements. I don't know!!!
One can construct an affine open cover of $V$ so that on each piece, $D$ is cut out by a single local equation $f$. Therefore it suffices to treat the problem in the case that $V=\operatorname{Spec} A$ is affine and $D$ is cut out by a single equation $f$.
As $D\subset W$, we have that $I_W \subset I_D \subset A$ where $I_W$ and $I_D$ are the ideals cutting out $W$ and $D$ respectively. But as $D$ is cut out by a single equation $f$, this means that $I_D=(f)$, so $I_W\subset (f)$. This exactly means that every element in $I_W$ can be written as $f\cdot r$ for some $r\in A$, and this is what the text means by dividing local equations.