the corresponding homotopy category, which is the universal solution to turning all weak equivalences into isomorphisms;

the corresponding $(\infty,1)$-category, which is, roughly, the universal solution to turning all weak equivalences into higher categorical equivalences. There are various versions of this construction depending on what model for $(\infty,1)$-categories is chosen.

Often, categories having weak equivalences also have extra structure that makes them easier to work with. A very powerful, and commonly occurring, level of such structure is called a model structure. There are also various weaker levels of structure, such as a category of fibrant objects.

A weak homotopy equivalence between topological spaces is a continuous function that induces (for all choices of basepoint) an isomorphism of all homotopy groups.