equality (definitional?, propositional, computational, judgemental, extensional, intensional, decidable)
isomorphism, weak equivalence, homotopy equivalence, weak homotopy equivalence, equivalence in an (∞,1)-category
Examples.
In general two objects are considered equivalent if they may be replaced by one another in all contexts under consideration, in particular whenever one is not doing anything evil.
At the propositional level, the general theory of equivalence is discussed at equivalence relation, but that leaves us with the question of the correct definition of equivalence in various situations. Furthermore, we can work on a higher level and ask what the equivalences are, not just whether things are equivalent.
In higher category theory an equivalence often means a morphism which is invertible in a maximally weak sense (that is, up to all higher equivalences). Two objects are equivalent if there is an equivalence between them. For example:
In a category (a $1$-category), an equivalence is an isomorphism.
In a (possibly weak) $2$-category, an equivalence is a morphism $f : x \to y$ such that there exists a morphism $g : y\to x$ and $2$-cell isomorphisms $f g \cong 1_y$ and $g f \cong 1_x$. In the $2$-category Cat this reduces to the usual notion of equivalence of categories (although one must define $\Cat$ carefully using anafunctors if one wants to avoid the axiom of choice).
In a (possibly weak) $3$-category, an equivalence is a morphism $f : x \to y$ such that there exists a morphism $g : y\to x$ and 2-cells $f g \to 1_y$ and $g f \to 1_x$ which are equivalences in the appropriate hom-2-categories. In the (weak) 3-category $Bicat$ of bicategories, pseudofunctors, pseudonatural transformations, and modifications, such equivalences are often called biequivalences, and that term is sometimes also used for an equivalence in any $3$-category.
In a (possibly weak) $\infty$-category, an equivalence is a morphism $f : x \to y$ such that there exists a morphism $g : y\to x$ and 2-cells $f g \to 1_y$ and $g f \to 1_x$ which are equivalences in the appropriate hom-$\infty$-categories. This can be made rigorous as a corecursive definition.
There is also the more structured/coherent notion of an adjoint equivalence, but one hopes to have a theorem that any equivalence can be improved to an adjoint one. This is known in some cases, including 2-categories, 3-categories, strict $\omega$-categories, and some models for $(\infty,1)$-categories.
In addition to this, there is the notion of equivalence of categories (and higher categories), which is on a separate page. But it is related: properly done, two $n$-categories are equivalent if and only if they're equivalent as objects in the $(n+1)$-category of $n$-categories.
In the category Top of topological spaces, notions weaker than isomorphism become very important: homotopy equivalence and weak homotopy equivalence. The latter notion is generalized to weak equivalence for objects in any model category. There is also a notion of equivalence between model categories: Quillen equivalence.
These can be understood to some extent using higher categories. For example, topological spaces should be weakly homotopy-equivalent if and only if they have equivalent fundamental infinity-groupoids. Similarly, Quillen equivalent model categories give rise to equivalent (infinity,1)-categories.
See equivalence in an (infinity,1)-category.
In homotopy type theory equivalences can be axiomatized as those terms of function types all whose homotopy fibers are contractible.
See equivalence in homotopy type theory.
equivalence, natural equivalence
homotopy equivalence, weak homotopy equivalence, Quillen equivalence