The Fenchel-Nielson coordinates are certain coordinates on Teichmüller space.
They parameterize Teichmüller space by cutting surfaces into pieces with geodesic boundaries and Euler characteristic $\xi = -1$. These building blocks (of hyperbolic 2d geometry) are precisely
the 3-holed sphere;
the 2-holed cusp;
the 1-holed 2-cusp;
the 3-cusp
Each surface of genus $g$ with $n$ marked points will have
$2g - 2 + n$ generalized pants;
$3 g - 3 + n$ closed curves.
The boundary lengths $\ell_i \in \mathbb{R}_+$ and twists $t_i \in \mathbb{R}$ of these pieces for
constitute the Fenchel-Nielsen coordinates on Teichmüller space $\Tau$.
Also use $\theta_i := t_i/\ell_i \in \mathbb{R}/\mathbb{Z}$
This constitutes is a real analytic atlas of Teichmüller space. On $M$ this reduces to coordinates $t_i \in \mathbb{R}/{\ell_i \mathbb{Z}}$, and these constitute a real analytic atlas of moduli space.