Every scheme $X$ over $S$ defines an algebraic space via its functor of points $h_X(T) = \mathrm{Hom}_S(T, X)$. The identity $X \to h_X$ is an étale atlas. This embedding $\mathrm{Sch}/S \hookrightarrow \mathrm{AlgSp}/S$ is fully faithful.

Let $X$ be a scheme and $G$ a finite group acting freely on $X$ (i.e., the map $G \times X \to X \times X$ given by $(g, x) \mapsto (gx, x)$ is a closed immersion). Define $R = G \times X$ with $s(g,x) = x$ and $t(g,x) = gx$. Then $R \rightrightarrows X$ is an étale equivalence relation, and the quotient $X/G$ is an algebraic space. When the action is not free, one must use algebraic stacks instead.

Consider $U = \mathbb{A}^1_k \sqcup \mathbb{A}^1_k$ and the equivalence relation $R$ that identifies the two copies away from the origin: $R = U \sqcup \mathbb{G}_{m,k}$ with appropriate source and target maps. The resulting algebraic space is the "line with doubled origin," which is an algebraic space but not a separated one. Indeed, it is also a scheme — a non-separated one.

Let $X = \mathbb{A}^2_k$ and $G = \mathbb{Z}/2\mathbb{Z}$ acting by $(x,y) \mapsto (-x, -y)$. The quotient $X/G$ exists as a scheme $\mathrm{Spec}\, k[x^2, xy, y^2] \cong \mathrm{Spec}\, k[a,b,c]/(ac - b^2)$. This is a quadric cone, and in this case the algebraic space is actually a scheme.

Let $k$ be an algebraically closed field and let $A$ be an abelian surface over $k$. Let $\iota: A \to A$ be the involution $a \mapsto -a$. The fixed locus consists of the 16 two-torsion points. Let $U = A \setminus \{0\}$ and $G = \{1, \iota|_U\}$. Since $\iota$ acts freely on $U$ (the other 15 fixed points being blown up), $U/G$ is an algebraic space. However, for suitable choices, one can arrange that this quotient is an algebraic space not isomorphic to any scheme.

Hironaka constructed a smooth proper threefold $X$ over $\mathbb{C}$ with a free $\mathbb{Z}/2\mathbb{Z}$-action such that the quotient $X / (\mathbb{Z}/2\mathbb{Z})$ is a proper smooth algebraic space that is not a scheme. This is one of the classical examples demonstrating the necessity of algebraic spaces. The key point is that $X$ is a proper variety, but the quotient, while proper as an algebraic space, does not admit an ample line bundle, and therefore is not projective and indeed not a scheme.

If $X$ is an algebraic space admitting an étale atlas $U \to X$ with $U$ affine, and if $X$ is quasi-separated, then $X$ is actually an affine scheme. More generally, every quasi-separated algebraic space that is affine over a scheme is itself a scheme.

Consider a scheme $S$ and an étale cover $\{U_i \to S\}$ with a descent datum for quasi-coherent sheaves that is effective. Now consider the moduli functor that parametrizes descent data. When descent fails to be effective (in the Zariski topology), the resulting functor may be an algebraic space rather than a scheme, witnessing the strictly larger scope of the étale topology.

For certain moduli problems, the Hilbert functor is representable by a scheme (by Grothendieck's theorem). However, relative Hilbert functors for algebraic spaces, $\mathrm{Hilb}_{X/S}$ where $X$ is an algebraic space over $S$, exist as algebraic spaces. This generalization is due to Artin.

Over $\mathbb{C}$, let $X$ be a smooth scheme and $G$ a finite group acting properly on $X^{\mathrm{an}}$. By GAGA-type results, if the quotient $X^{\mathrm{an}}/G$ is algebraizable, it carries the structure of an algebraic space. The passage from analytic to algebraic often forces one to work with algebraic spaces rather than schemes.

Let $S$ be a scheme, $X$ an $S$-scheme, and $G \to S$ an étale group scheme acting freely on $X$. Then the fppf quotient sheaf $X/G$ is an algebraic space over $S$. When $G$ is a constant finite group, this recovers the earlier example.

Let $L/K$ be a finite separable field extension, and let $X$ be a scheme over $L$. The Weil restriction $\mathrm{Res}_{L/K}(X)$ is defined as the functor $T \mapsto X(T \times_K L)$. When $X$ is quasi-projective, $\mathrm{Res}_{L/K}(X)$ is a scheme. For more general $X$, the Weil restriction exists as an algebraic space over $K$.

The moduli stack $\mathcal{M}_g$ of smooth curves of genus $g \geq 2$ is a smooth proper Deligne-Mumford stack. Its coarse moduli space $M_g$ is an algebraic space (and in fact a quasi-projective scheme by the Keel-Mori theorem combined with results on ampleness). For $g = 1$ with level structure, similar constructions produce algebraic spaces.