If $M,N$ and $P$ are $A$-modules then a mapping $f:M \times N\to P$ is bilinear if
1)-$f(ax+y,z)=af(x,z)+f(y,z)$ where $x,y\in M$ & $z\in N$ and $a\in A$
2)-$f(x,ay+z)=af(x,y)+f(x,z)$ where $x\in M$ & $y,z\in N$ and $a\in A$.
Can anyone give me some examples of modules $M,N$ & $P$ over $\mathbb{Z}$ and a bilinear mapping $f$ so that I can understand it better?
Choose any ring $R$ and consider the multiplication map $$R\times R\to R, (r,s)\mapsto rs.$$ This will give you a bilinear map on the underlying abelian groups.