Tightness/overtwistedness is pretty hard.  One place to start would be the paper <a href="http://arxiv.org/abs/math/0510639">Right-veering diffeomorphisms of compact surfaces with boundary I</a> by Honda-Kazez-Matic, which gives a necessary and sufficient condition in terms of "right-veering" mapping classes.  However, their result is not algorithmic since it requires checking something on *all* open books, not just a particular given one.  Andy Wand has recently claimed to have an algorithm that detects tightness.  I don't believe that there is a publicly available preprint yet, but there is an online <a href="http://scgp.stonybrook.edu/archives/5839">talk</a> by Wand and also some nice blog posts by Peter Lambert-Cole <a href="http://electrichandleslide.wordpress.com/2013/05/17/the-legendrian-surgery-conjecture/">here</a> and <a href="http://electrichandleslide.wordpress.com/2013/05/26/tightness-and-right-veering-monodromies/">here</a>.