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"?



Distributivity / commutativity of pushouts and pullbacks
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 D-shaped limits and C-shaped colimits. Then D-limits commute in A with C-shaped colimits when the functor 
$[D,A]\to A$ which calculates the limit preserves C-colimits. This is equivalent to saying 
that the functor $[C,A]\to A$ which calculates the colimit preserves D-limits. 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. 


