I come from a mathematics background and I'm familiar with laplace transforms in-as-far-as they provide an algebraic method to solve ordinary differential equations.
So, when I read about transfer functions, I can clearly see the one-to-one correspondence between the poles of a transfer function and the eigenvalues used to solve the homogeneous ODE. But I'm rather confused as to why the zeros of the transfer function are of particular interest to control engineers. If we're ultimately interested in stability of the system, why even bother with the zeros of the transfer function? The poles seem to tell you everything you really need to know. Does the location of a transfer function's zero with respect to its poles matter in some way? More importantly, how does their existence (or lack there of) affect how we design a suitable controller?