The simplest example: $\mathbb{A}^1_k \hookrightarrow \mathbb{P}^1_k$ is a Nagata compactification. The boundary is a single point $\{\infty\}$. This extends to $\mathbb{A}^n_k \hookrightarrow \mathbb{P}^n_k$ with boundary the hyperplane at infinity.

Let $X = V(f) \subset \mathbb{A}^n_k$ be an affine variety. Its projective closure $\overline{X} = V(f^h) \subset \mathbb{P}^n_k$ (where $f^h$ is the homogenization) is a Nagata compactification. However, $\overline{X}$ may be singular even if $X$ is smooth, in which case one might prefer a different compactification.

Let $\overline{X}$ be a proper scheme and $D \subset \overline{X}$ a closed subscheme. Then $X = \overline{X} \setminus D$ is a separated scheme of finite type, and $X \hookrightarrow \overline{X}$ is a compactification. Nagata's theorem says the converse: every such $X$ admits at least one such compactification.

The algebraic torus $\mathbb{G}_m^n = (\mathbb{A}^1 \setminus \{0\})^n$ can be compactified in many ways: $(\mathbb{P}^1)^n$, toric varieties associated to fans, or the projective space via the Segre embedding. Toric geometry provides a systematic framework for compactifying tori and torus orbits.

Let $C$ be a smooth affine curve over an algebraically closed field $k$. Then $C$ admits a unique smooth compactification $\overline{C}$: the smooth projective curve containing $C$ as a dense open subset. The boundary $\overline{C} \setminus C$ is a finite set of points. Uniqueness follows from the fact that birational smooth projective curves are isomorphic.

For a smooth surface $X$ over $k$, there exist smooth compactifications $\overline{X}$ (in characteristic zero, by Hironaka's resolution; in any characteristic for surfaces, by Lipman's resolution). The boundary can be arranged to be a simple normal crossing divisor.

Let $G$ be a finite group acting freely on a separated scheme $X$ of finite type over $S$. If $\overline{X}$ is a $G$-equivariant compactification of $X$ (i.e., the $G$-action extends to $\overline{X}$), then $\overline{X}/G$ is a compactification of $X/G$. However, finding equivariant compactifications requires extra work (it is a theorem of Sumihiro for torus actions and can be arranged for finite group actions by averaging).

Let $X$ be a separated algebraic space of finite type over a field that is not a scheme (e.g., Hironaka's quotient). Then Nagata's theorem provides a proper algebraic space $\overline{X}$ containing $X$ as a dense open subspace. By Chow's lemma, $\overline{X}$ is dominated by a projective scheme.

The moduli space $M_g$ of smooth genus-$g$ curves is a quasi-projective variety. Its Deligne-Mumford compactification $\overline{M}_g$ (parametrizing stable curves) is a Nagata compactification. The boundary $\overline{M}_g \setminus M_g$ parametrizes singular stable curves with nodes.

Let $S = \mathrm{Spec}(R)$ where $R$ is a DVR, and $X \to S$ a smooth separated family of curves over the generic fiber. Nagata's theorem gives a proper $\overline{X} \to S$, providing a (possibly singular) model over the special fiber. The semistable reduction theorem gives a compactification with mild singularities (nodes) after base change.

If $f: X \to S$ is not separated, Nagata's theorem fails. For instance, the line with doubled origin over $\mathrm{Spec}(k)$ does not embed as a dense open in any proper scheme (it is already "complete" in the non-separated sense). Separatedness is essential.

In characteristic zero, one can always find a smooth compactification of a smooth variety (by Hironaka). In positive characteristic, this is a major open problem in dimensions $\geq 4$. Nagata's theorem gives a compactification, but possibly singular.