Are you familiar with the notion of Kodaira dimension of a smooth variety $X$? If not, it is essentially the dimension of $X$ under the map associated to a sufficiently large power of the canonical bundle. Obviously this quantity is at most $\text{dim}\, X$, in which case $X$ is said to be of general type. Furthermore, this definition generalizes to singular varieties; the Kodaira dimension of $X$ singular is simply the Kodaira dimension of any resolution of singularities. I bring this all up because $M_g$ is of general type for $g\geq 23$ by work of Eisenbud-Harris-Mumford from the 80s, which implies the following: if $C$ is a general curve of genus $\geq 23$, and $S$ is a surface on which $C$ moves in a nontrivial linear system, then $S$ is birational to the product of $C$ with another curve. In other words, general curves do not occur as hyperplane sections of surfaces except in trivial ways.
In contrast, it is classical that for $g\leq 10$, $M_g$ is unirational, i.e. there is a dominant rational map $\mathbb A^n\dashrightarrow M_g$. That is to say, there is an open subset of an affine space whose points parametrize the general curve of genus $g$. THAT is to say, in these cases one can write down the equation of a general curve of genus $g$, as one represents the general cubic in Weierstrass form.
Edit: there are a few more things worth saying quickly. The moduli space of curves is separated, and what's more its compactification, the moduli space of stable curves $\overline{\mathcal{M}}_g$ is both separated and proper. What this means is that limits exist (properness) and are unique (separatedness), so if I have a family of smooth curves whose central fiber is either (a) missing or (b) in possession of nastier singularities than ordinary nodes, I can fill in/replace this fiber with a uniquely determined stable curve.
