equality (definitional, propositional, computational, judgemental, extensional, intensional, decidable)
isomorphism, weak equivalence, homotopy equivalence, weak homotopy equivalence, equivalence in an (∞,1)-category
Examples.
A weak equivalence is a morphism in a category $C$ which is supposed to be a true equivalence in a higher categorical refinement of $C$.
The bare minimum of axioms to be satisfied by a weak equivalence are encoded in the concepts of category with weak equivalences and homotopical category. For such categories one can consider
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.
The Dwyer-Kan localization uses simplicially enriched categories to model $(\infty,1)$-categories.
If we use complete Segal spaces or quasicategories to model $(\infty,1)$-categories, then the construction is a version of fibrant replacement.
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.
weak equivalence