We detected the 50 kyr old pulsar J1809-1917, and resolved its pulsar wind nebula (PWN) with the Chandra X-Ray Observatory. The pulsar spectrum fits a power-law+blackbody (PL+BB) model, with a photon index Γpsr=1.2±0.6 and T~2 MK for nH=0.7x1022/cm2. At a distance of 3.5 kpc, the luminosities of the PL and BB components are (4±1)x1031 ergs/s (in 0.5-8 keV) and ∼1x1032 ergs/s (bolometric), respectively. The bright inner PWN component of a 3"x12" size is elongated in the north-south direction (with the pulsar close to its south end) and immersed in a larger (20"x40"), similarly elongated outer PWN component. The PWN shape can be explained by the ram pressure confinement of the pulsar wind due to the supersonic motion of the pulsar. The PWN spectrum fits an absorbed PL model with nH~0.7x1022/cm2, Γpwn=1.4±0.1, and a 0.5-8 keV luminosity of ~4x1032 ergs/s. The compact PWN is surrounded by a large-scale (~4'x4') emission, more extended southward of the pulsar, i.e., in the direction of the alleged pulsar motion. To explain the extended X-ray emission ahead of the moving pulsar, one has to invoke strong intrinsic anisotropy of the pulsar wind, or assume that this emission comes from a relic PWN swept by the reverse SNR shock. The extension of the large-scale X-ray emission toward the center of the unidentified TeV source HESS J1809-193 suggests an association between this source and J1809-1917. An alternative source of relativistic electrons powering HESS J1809-193 might be the serendipitously discovered X-ray source CXOU J180940.7-192544. In addition to the cosmic microwave background radiation or starlight background, the seed photons for Compton up-scattering to TeV energies might be supplied by bright infrared emission from molecular/dust clouds seen within HESS J1809-193.