Lang Fiber Products STATEMENT: Let $\mathcal{C}$ be a category.A product in $\mathcal{C}_z$ is called the fiber product of $f$ and $g$ in $\mathcal{C}$ and is denoted by $X\times_zY$, together with its natural morphisms on $X,Y$, over $Z$, which are sometimes not denoted by anything.
QUESTION: What is $X\times_zY$. Is it just the standard cartesian product of $X$ and $Y$  ?

If $Z$ is a final object then the fibered product coincides with the usual product, however this is not true in general.  
Note that the definition of the fibered product also includes ''projections'' $\pi_X: X\times_Z Y\to X$ and $\pi_X: X\times_Z Y\to  Y$, as well as two morphisms $\alpha:X\to Z$ and $\beta:Y\to Z$ such that $\alpha \circ \pi_X = \beta\circ \pi_Y$.  The fibered product is also required to be universal and is thus defined uniquely up to unique isomorphism using the standard construction.
In the category Set for example we have $X\times_Z Y = \{(x,y)\in X\times Y \,|\, \alpha(x)=\beta(y)\}$.
Just thought I should add that the fibered product needn't look like a restricted Cartesian product.  Consider the case of the category of open sets in a given topology where the morphisms are just the inclusion maps.  In this case the fibered product of two sets is their intersection.
