A model category is a category equipped with notions of weak equivalences, fibrations and cofibrations allowing to run arguments similar to those of classical homotopy theory.
A model category is a category equipped with a model structure: three subcategories whose arrows are respectively called cofibrations, fibrations and weak equivalences satisfying a certain amount of lifting conditions.
Model categories are used to model the homotopy theory of certain objects, in particular to define derived functors and derived mapping spaces.
