  The first gravitational wave (GW) - gamma-ray burst (GRB) association,
GW170817/GRB 170817A, had an offset in time, with the GRB trigger time delayed
by $\sim$1.7 s with respect to the merger time of the GW signal. We generally
discuss the astrophysical origin of the delay time, $\Delta t$, of GW-GRB
associations within the context of compact binary coalescence (CBC) -- short
GRB (sGRB) associations and GW burst -- long GRB (lGRB) associations. In
general, the delay time should include three terms, the time to launch a clean
(relativistic) jet, $\Delta t_{\rm jet}$; the time for the jet to break out
from the surrounding medium, $\Delta t_{\rm bo}$; and the time for the jet to
reach the energy dissipation and GRB emission site, $\Delta t_{\rm GRB}$. For
CBC-sGRB associations, $\Delta t_{\rm jet}$ and $\Delta t_{\rm bo}$ are
correlated, and the final delay can be from 10 ms to a few seconds. For
GWB-lGRB associations, $\Delta t_{\rm jet}$ and $\Delta t_{\rm bo}$ are
independent. The latter is at least $\sim$10 s, so that $\Delta t$ of these
associations is at least this long. For certain jet launching mechanisms of
lGRBs, $\Delta t$ can be minutes or even hours long due to the extended engine
waiting time $\Delta t_{\rm wait}$ to launch a jet. We discuss the cases of
GW170817/GRB 170817A and GW150914/GW150914-GBM within this theoretical
framework and suggest that the delay times of future GW/GRB associations will
shed light into the jet launching mechanisms of GRBs.
