Let $X$ (resp. $Y$) be the affine $k$-scheme defined by the ideal $I$ (resp. $J$) in the polynomial ring $k[x\_1,...x\_n]$ (resp. $k[y\_1,...,y\_m]$).
Let $Z$ be the affine scheme defined by the ideal $L$ in $k[z\_1,...z\_s]$, and let $f^\\*:k[z]/L\rightarrow k[x]/I$ (resp. $g^\\*:k[z]/L\rightarrow k[y]/J$) be $k$-homomorphisms, where $x=(x\_1,...,x\_n)$ and so forth, corresponding to scheme morphisms $f:X\rightarrow Z$ (resp. $Y\rightarrow Z$).


Then it should be possible to express the fiber product $X\times\_{f,Z,g}Y$ via an ideal $W$ in the polinomial ring $k[x,y,z]$ [edit: actually, $W$ should be an ideal in $k[x,y]$] (where $x$ stands for the string of variables $x\_1,...,x\_n$, and so on).


Question: how to express $W\subseteq k[x,y,z]$ explicitely in terms of $I$, $J$, $L$, $f^\\*$ and $g^\\*$?


Edit:
You can express things explicitely in terms of some polynomials $F\_i$, $G\_i$ and $H\_i$ such that $I=(F\_1,...,F\_N)$, $J=(G\_1,...,G\_M)$ and $L=(H\_1,...,H\_S)$, and in terms of the components $(f\_1,...,f\_s)$ (resp. $(g\_1,...,g\_s)$) of $f$ (resp. $g$).



Expressing fiber product of affines via an ideal
Dear unknown, let me first congratulate you on the clearness of your question and the quality of your notation, which I'm now going to use.


The fibre product $X\times\_Z Y$ is the subscheme of $\mathbb A\_k^n \times \mathbb A\_k^m$ described by an ideal $\mathfrak A \subset k[x,y]$.
That ideal is $\mathfrak A=I^e + J^e + D$, where 


$I^e$ is the extension of $I\subset k[x]$ to $I^e\subset k[x,y]$,


$J^e$ is the extension of $J\subset k[y]$ to $J^e\subset k[x,y]$,


$D$ is the ideal generated by the $s$ differences $f\_i(x)-g\_i(y),\quad (i=1,\ldots,s)$ 


I find it clearer not to use generators for $I$ and $J$ and, strangely, $L$ is not used at all: this is because the fibre product is the same whether considered over $Z$ or over 
$\mathbb A\_k^s$ !


