Let $S_g$ be a compact Riemann surface of genus $g$ with $n$ marked points $x_1, \ldots, x_n$, and set $S_{g, \, n}:=S_g - \{x_0, \ldots, x_n\}$. Also, denote by $\pi_{g, \, n}$ the fundamental group of $S_{g, \, n}$ (we omit the decoration $n$ when $n=0$).

Then the mapping class group $\Gamma_{g,\, n}$ is defined as $$\Gamma_{g, \, n} := \pi_0 \, \textrm{Diff}^{+}(S_{g, \, n}),$$
where $\textrm{Diff}^{+}(S_{g, \, n})$ is the group of orientation-preserving diffeomorphisms of $S_g$ that fix each $x_i$.

The group $\Gamma_{g, \, n}$ naturally acts on the Teichmueller space $T_{g, \, n}$. Such an action is properly discontinuous and there is a subgroup of finite index acting freely, in such a way that, as an orbifold, $$M_{g, \, n}= T_{g, \, n}/ \Gamma_{g, \, n}.$$
In particular, $\pi^{\textrm orb}_1(M_{g, \, n}) = \Gamma_{g, \, n}$.

By a result of Baer and Nielsen, with this representation $\Gamma_{g}$ is identified with the subgroup of $\textrm{Out}(\pi_{g})$ acting trivially on $H_2(\pi_{g})$. When $n \geq 1$, there are similar characterizations.

For (many) more details, see the paper

R. Hain, E. Looijenga: *Mapping class groups and moduli spaces of curves,* Algebraic Geometry-Santa Cruz 1995, 97–142, Proc. Sympos. Pure Math., **62**, Part 2, Amer. Math. Soc., Providence, RI (1997),

also available on the arXiv.