The category $\mathbf{Vect}_k^{\mathrm{fd}}$ of finite-dimensional vector spaces over $k$ is equivalent to $\mathbf{Mat}_k$ (objects: natural numbers, morphisms: matrices). The equivalence sends $V$ to $\dim V$ and a linear map to its matrix with respect to chosen bases. The skeleton of $\mathbf{Vect}_k^{\mathrm{fd}}$ is $\mathbf{Mat}_k$: each $n$-dimensional space is isomorphic to $k^n$.

Gelfand duality: the category of compact Hausdorff spaces is equivalent to the opposite of the category of commutative unital C*-algebras. The functors are $X \mapsto C(X)$ (continuous functions) and $A \mapsto \mathrm{Max}(A)$ (maximal ideal space).

The category of affine schemes is equivalent to $\mathbf{CRing}^{\mathrm{op}}$. The equivalence sends $\mathrm{Spec}\, A \mapsto A$ and $A \mapsto \mathrm{Spec}\, A$. This is the starting point of algebraic geometry: geometric objects (affine schemes) correspond to algebraic objects (commutative rings) contravariantly.

For a connected, locally simply connected space $X$ with fundamental group $\pi_1(X, x_0) = G$, the category of covering spaces of $X$ is equivalent to the category of $G$-sets: $\mathbf{Cov}(X) \simeq G\text{-}\mathbf{Set}$. The equivalence sends a covering $p : \tilde{X} \to X$ to the fiber $p^{-1}(x_0)$ with the monodromy action.

For a discrete category with objects $\{1, 2, \ldots, n\}$, the presheaf category $[\mathcal{C}^{\mathrm{op}}, \mathbf{Set}]$ is equivalent to $\mathbf{Set}^n$ (the category of $n$-tuples of sets). A presheaf on a discrete category is simply a choice of one set for each object.

Two rings $R$ and $S$ are Morita equivalent if their module categories are equivalent: $R\text{-}\mathbf{Mod} \simeq S\text{-}\mathbf{Mod}$. For example, $R$ and $M_n(R)$ are always Morita equivalent. Morita equivalence preserves all categorical properties of module categories (projectivity, injectivity, exactness) but not ring-theoretic properties (commutativity, being a field).

For a group $G$, the category of $G$-sets is equivalent to the functor category $[\mathbf{B}G, \mathbf{Set}]$, where $\mathbf{B}G$ is the one-object category with morphisms $G$. A functor $\mathbf{B}G \to \mathbf{Set}$ picks a set $S$ and gives an action $G \times S \to S$; natural transformations correspond to equivariant maps.

Let $\mathbf{FinSet}$ be the category of finite sets and let $\mathbf{F}$ be the category whose objects are $\{[n] = \{1, \ldots, n\} : n \geq 0\}$ (with $[0] = \varnothing$) and whose morphisms are all functions. Then $\mathbf{F}$ is a skeleton of $\mathbf{FinSet}$: every finite set is isomorphic to exactly one $[n]$.

A connected groupoid (a groupoid with a single isomorphism class) is equivalent to $\mathbf{B}G$ for some group $G$: pick any object $x$, set $G = \mathrm{Aut}(x)$, and the equivalence $\mathbf{B}G \to \mathcal{G}$ sends $*$ to $x$. In particular, the fundamental groupoid $\Pi_1(X)$ of a path-connected space $X$ is equivalent to $\mathbf{B}\pi_1(X, x_0)$.

$\mathbf{Set}$ and $\mathbf{Set}^{\mathrm{op}}$ are not equivalent. One way to see this: $\mathbf{Set}$ has a zero object candidate (the empty set is initial but not terminal), while in $\mathbf{Set}^{\mathrm{op}}$ the empty set is terminal but not initial. More precisely, $\mathbf{Set}$ has a strict initial object (no morphisms into $\varnothing$ from nonempty sets), but $\mathbf{Set}^{\mathrm{op}}$ does not have this property for its initial object.

If $\mathcal{C} \simeq \mathcal{D}$, then $\mathcal{C}$ has all (co)limits of a given type if and only if $\mathcal{D}$ does. For example, $\mathbf{Ab}$ is an abelian category, so any category equivalent to $\mathbf{Ab}$ is also abelian. Equivalence also preserves: having enough injectives/projectives, being locally small, having a generator, etc.

In the Gelfand-Manin framework, a key question is when two abelian categories $\mathcal{A}$ and $\mathcal{B}$ have equivalent derived categories $D^b(\mathcal{A}) \simeq D^b(\mathcal{B})$. This is a weaker condition than $\mathcal{A} \simeq \mathcal{B}$ and is the subject of Fourier-Mukai theory and tilting theory. For instance, $D^b(\mathrm{Coh}(X)) \simeq D^b(\mathrm{Coh}(Y))$ for certain non-isomorphic varieties $X$ and $Y$.