We present a 100 ks observation of the Galactic supernova remnant 3C 396 (G39.2-0.3) with the Chandra X-Ray Observatory that we compare with a 20 cm map of the remnant from the Very Large Array. In the Chandra images, a nonthermal nebula containing an embedded pointlike source is apparent near the center of the remnant that we interpret as a synchrotron pulsar wind nebula surrounding a yet undetected pulsar. From the 2-10 keV spectrum for the nebula [N-H = (5.3 +/- 0.9) x 10(22) cm(-2), Gamma = 1.5 +/- 0.3] we derive an unabsorbed X-ray flux of S-X = 1.62 x 10(-12) ergs cm(-2) s(-1), and from this we estimate the spin-down power of the neutron star to be (E)over dot = 7.2 x 10(36) ergs s(-1). The central nebula is morphologically complex, showing bent, extended structure. The radio and X- ray shells of the remnant correlate poorly on large scales, particularly on the eastern half of the remnant, which appears very faint in X-ray images. At both radio and X-ray wavelengths, the western half of the remnant is substantially brighter than the eastern half.