Belyi's theorem states that the following properties of a nonsingular projective algebraic curve $X$ are equivalent:

1) $X$ is defined over $\overline{\mathbb{Q}};$

2) There exists a meromorphic function $\phi: X\to\mathbb{P}^1\mathbb{C} $ ramified at most at $0,1,$ and $\infty$;

3) $X$ is isomorphic to $\Gamma \backslash \mathbb{H}$ (compactified at cusps) for a finite index subgroup $[\mathrm{PSL}_2(\mathbb{Z}), \Gamma]<\infty.$

The remaining question is: $$\boxed{\text{ Is there a way to treat singularities in this or a similar framework? }}$$

The following of my original questions have been answered:

Can this be generalized to arbitrary projective nonsingular varities of higher dimensions? (I discussed this with one professor here in Goettingen. That seems to be ongoing research. Please see also comment of David Roberts.)

What compactification do they mean here? $\Gamma \backslash \mathbb{H} \cup \mathbb{Q}$! (see the answer of Robin Chapman)

What is a nice reference for the proof of Belyi's theorem? (see answer of YBL and Koeck + the comments of Emerton)

How does the Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ enter the picture? (see comment of Ariyan and answer of YBL)

Where can I find nice examples where these computations have been done explicitly? (see answer of Andy Putnam and JSE)