Using a minimum of technical vocabulary, give a summary of why it is that the moduli space of genus g complex curves with n marked points has a natural compactification that is isomorphic (as a complex orbifold) to a projective algebraic variety.

The classical (pre DeligneMumford) approach is to map $\mathcal{M}_g$ into $\mathcal{A}_g$ using the Torreli map. Whereas the classical way to see that $\mathcal{A}_g$ is quasiprojective is to define it as the Siegel upper half space $\mathcal{H}_g$ ($g$ by $g$ complex matrices with positive define imaginary part), modulu $\mathrm{SP}(2g,\mathbb{Z})$; We define the level cover $\mathcal{A}_g(m)\to\mathcal{A}_g$, which is the moduli of $\mathcal{A}_g$ plus torsion points, which is the quotient of $\mathcal{H}_g$ by $\Gamma(m)$ (the matrixes in $\mathrm{SP}(2g,\mathbb{Z})$ which are trivial modulo $n$), and send $\mathcal{A}_g(m)$ to some projective space using polynomials in the theta constants. Reference (for both Torreli and the embedding of $\mathcal{A}_g$ above) : Mumfords curves and their Jacobians (now bundled together with the red book in LNM 1358) Lecture IV. In Lecture II (same place) Mumford sketches two more "coordinate oriented" methods
He also says these are the only "coordinate oriented" methods he knows of. 


I'm not sure how useful this will be, but I can say a couple of words about why A_g is quasiprojective (I am very much not an expert, so real experts are welcome to correct me). This will be an outline of what happens in the proof, but without going into the details I can't explain why this works. There are probably fancier ways of doing this, but the most classical is via the theta constants. The best complete reference for this is Igusa's book "Theta Functions", but a large part of the story can also be found in Mumford's "Tata Lectures on Theta I", which is easier to read. In particular, the 1dimensional case is worked out there very explicitly, and if you want to understand the details I would try to understand this case first. By the way, in the 1dimensional case we have A_{g} = M_{g} since every elliptic curve is isomorphic to its Jacobian. The basic idea is that theta functions provide a natural set of coordinates on A_g. These coordinates give an embedding into projective space, and the compactification is the closure of this embedding. A point of A_g is a complex tori T that happens to be a projective variety (together with a principal polarization, but don't worry about that now). A theta function on T is a holomorphic function on C^g (the universal cover of T) that satisfies a simple transformation law (it is really a section of a certain bundle on T lifted to the universal cover C^g). The set of all theta functions on T spans a finitedimensional vector space, and you can use them to embed T into projective space. Our goal, though, is to embed A_g into projective space. The universal cover of A_g is the Siegel upper half plane H_g, which is a nice complex domain. In fact, it is the set of all skewsymmetric gxg complex matrices whose imaginary part is positive definite (for g=1, this is exactly the classical upper half plane). One now looks at the set of all holomorphic functions F(.,.) on H_g x C^g such that for all T in H_g, the function F(T,.) is a theta function on T. This again spans a finitedimensional vector space. Every PPAV has a distinguished point, namely 0. The "theta constants" or "theta nulls" are the functions F(.,0) on H_g. These satisfy a (somewhat complicated) transformation law on H_g coming from the fact that they are sections of a certain bundle on A_g. These thetanulls give you an embedding into projective space, and thus a compactification. 


Technically I recall from the Tata lectures, that it is Ag(4,8) that embeds via theta nulls. Another nice reference is Freitag's book on Siegel modular functions. 


The moduli space $\overline{M}_{g,n}$ has a structure of orbifold or in algbraic terms of DeligneMumford algebraic stack. Since the deformations of $n$pointed genus $g$ DeligneMumford stable curves (at most nodal singularities and finite automorphism group) are unobstructed the stack is smooth of dimension $3g3+n$. 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. 

