The moduli space $\overline{M}_{g,n}$ has a structure of orbifold or in algbraic terms of Deligne-Mumford algebraic stack.
Since the deformations of $n$-pointed genus $g$ Deligne-Mumford stable curves (at most nodal singularities and finite automorphism group) are unobstructed the stack is smooth of dimension $3g-3+n$.
The corresponding coarse moduli space $\overline{M}_{g,n}$ is a projective variety with quotient singularities at the places where the automorphisms groups of the curves jump.
To understand why it is projective one can consider the usual GIT construction. One can embed a curve of genus $g\geq 2$ in $\mathbb{P}^{N}$ with the sections of the $3$-canonical system. The action of $SL(N)$ on the Hilbert scheme $H$ of such curves can be linearized. Then one construct $\overline{M}_{g}$ as $H/SL(N)$. Now the projectivity follows from the projectivity of $H$ and standard theorems of GIT.