  We present a deep observation with the Chandra X-ray Observatory of the
neutron star bow shock G189.22+2.90 in the supernova remnant (SNR) IC 443. Our
data confirm the cometary morphology and central point source seen previously,
but also reveal considerable new structure. Specifically, we find that the
X-ray nebula consists of two distinct components: a "tongue" of bright emission
close to the neutron star, enveloped by a larger, fainter "tail". We interpret
the tongue and tail as delineating the termination shock and the post-shock
flow, respectively, as previously identified also in the pulsar bow shock
G359.23-0.82 ("the Mouse"). However, for G189.22+2.90 the tongue is much less
elongated than for the Mouse, while the tail is much broader. These differences
are consistent with the low Mach number, M >~ 2, expected for a neutron star
moving through the hot gas in a SNR's interior, supporting the case for a
physical association between G189.22+2.90 and IC 443. We resolve the stand-off
distance between the star and the head of the bow shock, which allows us to
estimate a space velocity for the neutron star of ~230 km/s, independent of
distance. We detect thermal emission from the neutron star surface at a
temperature of 102 +/- 22 eV, which is consistent with the age of SNR IC 443
for standard neutron star cooling models. We also identify two compact knots of
hard emission located 1-2 arcsec north and south of the neutron star.
