What does it exactly mean to say that in a certain category pushouts and pullbacks "commute"? Is it the same to say that they "distribute"?
The condition Martin and Todd mention is indeed a sort of distributivity condition. It is also often called {\em stability} of pushouts. I think that it should not be called commutativity. Let D and C be small categories, and A a category with Dshaped limits and Cshaped colimits. Then Dlimits commute in A with Cshaped colimits when the functor $[D,A]\to A$ which calculates the limit preserves Ccolimits. This is equivalent to saying that the functor $[C,A]\to A$ which calculates the colimit preserves Dlimits. The most famous example is commutativity of finite limits with filtered colimits. Note, however, that commutativity of pullbacks and pushouts in this sense is rare, and is probably not what was meant. 


I think one would need some context to be sure how this is meant to be interpreted, but one possible interpretation would be subsumed under the condition that a pullback functor between slice categories preserves colimits. So: suppose we have a pushout $h: B \to P \leftarrow C: j$ of a diagram $$B \stackrel{f}{\leftarrow} A \stackrel{g}{\to} C$$ and suppose given an arrow $k: Q \to P$. Then you can pull back the pushout diagram targeted at $P$ along $k$ to get a square terminating at $Q$. If this square is a pushout, then you could say that the pullback functor $k^\ast$ preserves the pushout. This happens for example in toposes $E$: every pullback functor $$k^\ast: E/P \to E/Q$$ preserves colimits. (Beaten by about a minute by Martin!) 


Basically all compatibility conditions may be phrased in form of commutativity; in particlar distributivity. Pushouts commute with pullbacks, if the following holds: For all morphisms $B \to A,C,D \to R$ the canonical morphism $(A\times_R D) \coprod_B (C \times_R D) \to (A \coprod_B C) \times_R D$ is an isomorphism. And indeed this looks like the distributivity law $AD+CD = (A+C)D$. 

