definition of a model category In Quillen's book ,,Homotopical Algebra" he defines a model category as a category with three types of arrows satyfying some axioms.
I have problems with understanding two of them. These axioms says among other things that fibrations are stable under base-change and cofibrations are stable under co-base change.
I would be thankfull if anyone could explain what exactly means this (co)base change and (co)base extension that also appers in the axioms.
 A: Do you mean the definition of (co)base change or the intuitive meaning of the axioms? 
A base change diagram is often called a pullback. This is a commutative square
$$(\ddagger)\qquad\begin{array}{ccc} Z & \stackrel{\scriptsize s}{\to} & X \\ {\scriptsize r}\downarrow && \downarrow{\scriptsize g} \\ Y & \stackrel{\scriptsize f}{\to } & A\end{array}$$ initial among all those having $X\to A\leftarrow Y$ fixed - see the link or any category theory book for details.
A class ${\mathscr E}$ of morphisms called stable under base change if in any pullback $(\ddagger)$, $g\in{\mathscr E}$ implies ${\mathscr r}\in{\mathscr E}$.
Dually, a cobase change diagram is often called a pushout, and ${\mathscr E}$ is called stable under cobase change if in a pushout diagram of the form $(\ddagger)$, $r\in{\mathscr E}$ implies $g\in{\mathscr E}$.
In model categories, classes of morphisms are often defined by their lifting properties with respect to other classes: If ${\mathscr C}$ is some class of morphisms, you can consider the class ${\mathscr C}^{\perp}$ of morphisms $g: X\to A$ having the property that in any commutative diagram $(\ddagger)$, $r\in {\mathscr C}$ implies that there exists a lift $Y\to X$ making everything commute. You can then check that ${\mathscr C}^{\perp}$ is always stable under base change / pullback - that's a good exercise on the universal property of the pullback. Similarly, you can define ${^{\perp}}{\mathscr C}$ as the class of morphisms $r :Z\to Y$ such that for any commutative diagram $(\ddagger)$, $g\in{\mathscr C}$ implies the existence of a lift $Y\to X$; then, ${^{\perp}}{\mathscr C}$ will always be stable under cobase change / pushout.
