Let $\Gamma$ be a Fuchsian group of signature $[g;m_1,\dots,m_r;s]$. When we quotient $\mathbb{H}^2$ by $\Gamma$ we obtain a genus $g$ surface with $s$ cusps and $r$ elliptic points with orders $m_1,\dots,m_r$.
Now, the concepts of genus and cusp have a geometric description i.e handles/holes and well, cusps. What is the equivalent description of an elliptic point and what does it look like?
They are sometimes called cone points, because they look like taking a piece of paper with a corner of angle $2\pi/m_i$ and folding it over to identify opposite sides of the vertex. This produces something which looks like a cone at the singular point. They are also sometimes called pillowcase points as they look like the corners of a pillow.
So, for example, if $m_i=2$ then take a piece of paper, pick a point on one side, and identify the sides opposite the point to make a cone. If $m_i=3$ take a piece with a $120^\circ$ angle and identify the opposite sides. The cone gets "sharper" as the angle decreases, i.e. as $m_i$ increases.