In the nerve $N(\mathcal{C})$ of an ordinary category, a $1$-simplex $f$ is an equivalence if and only if $f$ is an isomorphism in $\mathcal{C}$. There is no distinction between "equivalence" and "isomorphism" for ordinary categories.

In a Kan complex $X$ (an $\infty$-groupoid), every morphism (every $1$-simplex) is an equivalence. This is because outer horn fillers provide homotopy inverses: given $f: x \to y$, the outer horn $\Lambda^2_0$ with $d_1 = f$ and $d_2 = \mathrm{id}_x$ can be filled, producing $g: y \to x$ with $g \circ f \simeq \mathrm{id}_x$.

In the derived $\infty$-category $D(R)$ of a ring $R$, a morphism $f: C \to D$ of chain complexes is an equivalence if and only if $f$ is a quasi-isomorphism. The homotopy inverse is provided by the mapping cone construction and the long exact sequence in homology.

In the $\infty$-category $\mathcal{S}$ of spaces, a map $f: X \to Y$ is an equivalence if and only if $f$ is a weak homotopy equivalence. This is the $\infty$-categorical incarnation of Whitehead's theorem (for Kan complexes).

For $N(\mathcal{C})$, the core $N(\mathcal{C})^\simeq = N(\mathcal{C}^{\cong})$ is the nerve of the maximal subgroupoid of $\mathcal{C}$ (the groupoid of objects and isomorphisms).

For a Kan complex $X$, the core is $X$ itself: $X^\simeq = X$, since every morphism is already an equivalence.

The core of the derived $\infty$-category $D(R)^\simeq$ is the $\infty$-groupoid of chain complexes and quasi-isomorphisms between them. This is a "space" that classifies chain complexes up to quasi-isomorphism.

A functor $F: \mathcal{C} \to \mathcal{D}$ of ordinary categories induces a categorical equivalence $N(F): N(\mathcal{C}) \to N(\mathcal{D})$ if and only if $F$ is an equivalence of categories (fully faithful and essentially surjective).

If $\mathcal{M}$ is a model category with weak equivalences $W$, the localization map $N(\mathcal{M}) \to N(\mathcal{M})[W^{-1}]$ is a categorical equivalence when restricted to bifibrant objects. More precisely, the inclusion of bifibrant objects $\mathcal{M}^{cf} \hookrightarrow \mathcal{M}$ induces an equivalence $N(\mathcal{M}^{cf})[W^{-1}] \xrightarrow{\sim} N(\mathcal{M})[W^{-1}]$.

A categorical equivalence $F: \mathcal{C} \to \mathcal{D}$ between Kan complexes is the same as a weak homotopy equivalence (since both are fibrant in both the Joyal and Kan--Quillen model structures).

For general quasi-categories, categorical equivalences are strictly weaker than Kan--Quillen weak equivalences. For instance, the inclusion $\{0\} \hookrightarrow J$ (where $J$ is the walking isomorphism) is a categorical equivalence but not a Kan--Quillen weak equivalence (since $|J| \simeq [0,1] \not\simeq \{0\}$... actually $J$ is contractible, so this is a bad example). A better distinction: $\Delta[0] \to N(B\mathbb{Z})$ is a Kan--Quillen weak equivalence iff $B\mathbb{Z}$ is contractible (it is not: $\pi_1 = \mathbb{Z}$), so these two notions genuinely differ.

For a Kan complex $X$, the homotopy category $\operatorname{h}X$ is the fundamental groupoid $\Pi_1(X)$: objects are points, morphisms are homotopy classes of paths. This is a groupoid because every morphism in a Kan complex is an equivalence.

The homotopy category $\operatorname{h}\mathcal{S}$ of the $\infty$-category of spaces is equivalent to the classical homotopy category $\operatorname{Ho}(\mathbf{Top})$. Morphisms are homotopy classes of continuous maps between CW complexes.

$\operatorname{h}D(R)$ is the classical derived category as an ordinary triangulated category.