Geometric Intuition for "Right-Veering" Property of $f$ in MCG(S)? let $S$ be a compact surface with non-empty boundary, let $\alpha : [0,1] \rightarrow S$ be a Properly-embedded arc (meaning both  endpoints of the arc are in $\partial S$)  and let $f$ be an element in $MCG(S)$, the mapping class group of $S$. Let $\alpha(0)=x$. I am trying to understand the geometric motivation behind the property of "right-veeringness" of $f$. We say that $f$ is right-veering if the orientation given at the point $x$ by the pair $(f'(\alpha(0)) ,\alpha'(0))$ agrees with the orientation of the surface $S$ at the point $x$. Any ideas ?

For the overall definition: You need to either work in the universal cover and homotop lifts of $\alpha$ and $f(\alpha)$ so they don't intersect before checking the condition or work on the surface and homotop them so they intersect minimally. Then, $f$ is right veering if for all $\alpha$ the condition is true.
The condition: The condition is that the arc to $f(\alpha)$ starts out to the right of $\alpha$ at the boundary (i.e. at $0$). (That is, the initial tangent directions $(f \circ \alpha)'(0)$ and $\alpha'(0)$ form an oriented basis.) But this isn't robust under homotopy (or rather, isotopy of $f$ fixing the boundary), so the correct condition is to first homotop $f(\alpha)$ to intersect $\alpha$ minimally, and then see if it starts out heading rightward. (It would have to come across the arc an extra time if you change which direction it starts, so this is a well-defined concept.)

This first showed up in Honda-Kazez-Matic relating to tight contact structures. The basic idea is that $f$ not right-veering allows you to find an overtwisted disk in the contact manifold associated to the open book with monodromy $f$. Very roughly, this is because the way the contact structure associated to the open book works involves "continuing to twist further" in the right-veering way as you go closer to the binding. So non-right-veering means there'd have to be "extra twisting" and an overtwisted disk, meaning the contact structure is not tight.
