A model category is left proper if a pushout of a WE under a cofibration is always a WE. Dually, a model category is right proper if a pullback of a WE under a fibration is always a WE. A model category is proper if it is both left proper and right proper.
http://ncatlab.org/nlab/show/proper+model+category
http://golem.ph.utexas.edu/category/2012/05/the_mysterious_nature_of_right.html
Examples of proper model cats: , and simplicial abelian groups.
nLab page on Proper model category