  We present an observation with the Chandra X-ray Observatory of the unusual
radio source G359.23-0.82 ("the Mouse"), along with updated radio timing data
from the Parkes radio telescope on the coincident young pulsar J1747-2958. We
find that G359.23-0.82 is a very luminous X-ray source (L_X [0.5-8.0 keV] =
5e34 ergs/s for a distance of 5 kpc), whose morphology consists of a bright
head coincident with PSR J1747-2958, plus a 45"-long narrow tail whose
power-law spectrum steepens with distance from the pulsar. We thus confirm that
G359.23-0.82 is a bow-shock pulsar wind nebula powered by PSR J1747-2958; the
nebular stand-off distance implies that the pulsar is moving with a Mach number
of ~60, suggesting a space velocity ~600 km/s through gas of density ~0.3
cm^-3. We combine the theory of ion-dominated pulsar winds with hydrodynamic
simulations of pulsar bow shocks to show that a bright elongated X-ray and
radio feature extending 10'' behind the pulsar represents the surface of the
wind termination shock. The X-ray and radio "trails" seen in other pulsar bow
shocks may similarly represent the surface of the termination shock, rather
than particles in the postshock flow as is usually argued. The tail of the
Mouse contains two components: a relatively broad region seen only at radio
wavelengths, and a narrow region seen in both radio and X-rays. We propose that
the former represents material flowing from the wind shock ahead of the
pulsar's motion, while the latter corresponds to more weakly magnetized
material streaming from the backward termination shock. This study represents
the first consistent attempt to apply our understanding of "Crab-like" nebulae
to the growing group of bow shocks around high-velocity pulsars.
