We report the X-ray Multimirror Mission-Newton European Photon Imaging Camera observation of two elongated parallel x-ray tails trailing the pulsar Geminga. They are aligned with the object's supersonic motion, extend for ~2', and have a nonthermal spectrum produced by electron-synchrotron emission in the bow shock between the pulsar wind and the surrounding medium. Electron lifetime against synchrotron cooling matches the source transit time over the x-ray features' length. Such an x-ray detection of a pulsar bow shock (with no Hα emission) allows us to gauge the pulsar electron injection energy and the shock magnetic field while constraining the angle of Geminga's motion and the local matter density.