Definition about moduli space of Riemann Surfaces of genus $g$ with $n$ marked points, $\mathcal{M}_{g,n}$

Question

In the definition of the moduli space of Riemann Surfaces of genus $g$ with $n$ marked points, $\mathcal{M}_{g,n}$, it is asked that $g$ and $n$ satisfy the condition $2-2g-n<0$. I've seen that this means that the Compact Riemann Surface of genus $g$ with $n$ marked points, $\Sigma_{g,n}$, has negative Euler Characteristic. My questions are:

1. What does it mean that a Surface has a negative Euler characteristic?
2. Why do we need that condition?

Thank you in advance!