The trivial equivalence relation on $U$ is $R = U$ (the diagonal). The quotient is $U/U \cong U$ itself. This shows every scheme arises from an étale equivalence relation.

Let $G$ be a finite group acting freely on a scheme $U$ over a field $k$. Define $R = G \times U$ with $s(g, u) = u$ and $t(g, u) = g \cdot u$. Since $G$ is étale over $k$ (being a finite discrete group scheme in characteristic not dividing $|G|$), the projections $s, t: R \to U$ are étale. Freeness ensures $R \hookrightarrow U \times U$ is a monomorphism. The quotient $U/R = U/G$ is an algebraic space.

Let $U = \mathbb{A}^1_k$ and $G = \mathbb{Z}/n\mathbb{Z}$ acting by $x \mapsto x + 1$ (assuming $\mathrm{char}(k) \nmid n$ and $k$ contains a primitive $n$-th root of unity is not needed here; we only need $n \neq 0$ in $k$). Set $R = G \times U$. The quotient $\mathbb{A}^1_k / (\mathbb{Z}/n\mathbb{Z})$ is an algebraic space. When $n$ is invertible in $k$, this is isomorphic to $\mathbb{A}^1_k$ via the map $x \mapsto \prod_{i=0}^{n-1}(x - i)$.

Let $f: U \to X$ be an étale surjection of schemes. Then $R = U \times_X U \rightrightarrows U$ is an étale equivalence relation, and $U/R \cong X$. This provides a tautological presentation of any scheme as a quotient.

Let $X = \mathbb{P}^1_k \times \mathbb{P}^1_k$ and $\sigma$ the involution swapping the two factors. On the open set $U = X \setminus \Delta$ (complement of the diagonal), $\sigma$ acts freely. The quotient $U/\langle \sigma \rangle$ is an algebraic space (and in this case a scheme, since $U$ is quasi-projective).

Take $U = \mathbb{A}^1_k \sqcup \mathbb{A}^1_k$ and let $R \subset U \times U$ be the equivalence relation that identifies the two copies everywhere except at the origin. The maps $s, t: R \to U$ are étale (they are open immersions on the gluing part). The quotient is the line with doubled origin, a non-separated algebraic space (which happens to be a scheme).

Let $L/K$ be a finite Galois extension with group $G = \mathrm{Gal}(L/K)$. Let $U = \mathrm{Spec}(L)$. Then $R = \mathrm{Spec}(L \otimes_K L) \cong \coprod_{g \in G} \mathrm{Spec}(L)$ defines an étale equivalence relation, and $U/R \cong \mathrm{Spec}(K)$. This is Galois descent in its simplest form.

Let $U_1, U_2$ be schemes with an open subscheme $V_i \subset U_i$ and an étale isomorphism $\varphi: V_1 \xrightarrow{\sim} V_2$. Set $U = U_1 \sqcup U_2$ and $R = U \sqcup V_1$ (with appropriate maps using $\varphi$). The quotient $U/R$ is the algebraic space obtained by gluing $U_1$ and $U_2$ along $\varphi$. If $\varphi$ is not an open immersion on both sides, the result may fail to be a scheme.

Let $A$ be an abelian variety over $k$ and $H \subset A$ a finite étale subgroup scheme. Then $H$ acts freely on $A$ by translation: $R = H \times A \rightrightarrows A$. The quotient $A/H$ is an abelian variety (hence a scheme), and the map $A \to A/H$ is an isogeny. This is a case where the algebraic space quotient turns out to be a scheme.

Let $X = (\mathbb{P}^1)^n$ and $G = S_n$ (the symmetric group) acting by permuting factors. The quotient $X/S_n$ is the $n$-th symmetric product $\mathrm{Sym}^n(\mathbb{P}^1) \cong \mathbb{P}^n$ (a scheme). The étale equivalence relation $R = S_n \times X$ on the open locus of distinct points gives an étale presentation of the quotient.

Consider $U = \mathrm{Spec}(k[x,y])$ and the equivalence relation given by the free $\mathbb{Z}/2\mathbb{Z}$-action $(x,y) \mapsto (-x, -y)$ restricted to $U \setminus \{0\}$. The Zariski quotient does not exist as a scheme with good properties, but in the étale topology, $U \setminus \{0\} / (\mathbb{Z}/2\mathbb{Z})$ is a well-defined algebraic space.

Let $C$ be a smooth projective curve of genus $g \geq 2$ over $k$, and $\sigma: C \to C$ a fixed-point-free involution (which exists when $C$ admits a degree-2 map to a curve $C'$ unramified everywhere). Then $R = C \sqcup C$ with the identity and $\sigma$ gives an étale equivalence relation, and $C/R \cong C'$ is a curve of genus $(g+1)/2$ when $g$ is odd.

For any scheme $U$, the diagonal $\Delta_U: U \hookrightarrow U \times U$ is an étale equivalence relation (in fact a closed immersion if $U$ is separated). The quotient is $U$ itself. This trivial example illustrates that the notion of étale equivalence relation subsumes the notion of identity.