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We report on an observation of the Vela-like pulsar B1823-13 and itssynchrotron nebula with Chandra.The pulsar's spectrum fits a power-law modelwith a photon index Gamma_PSR=2.4 for the plausible hydrogen column densityn_H=10^{22} cm^{-2}, corresponding to the luminosity L_PSR=8*10^{31} ergss^{-1} in the 0.5-8 keV band, at a distance of 4 kpc. The pulsar radiationlikely includes magnetospheric and thermal components, but they cannot bereliably separated because of the small number of counts detected and stronginterstellar absorption. The pulsar is surrounded by a compact, 25''x 10'',pulsar wind nebula (PWN) elongated in the east-west direction, which includes abrighter inner component, 7''x 3'', elongated in the northeast-southwestdirection. The slope of the compact PWN spectrum is Gamma_comp=1.3, and the0.5-8 keV luminosity is L_comp~3*10^{32} ergs s^{-1}. The compact PWN issurrounded by asymmetric diffuse emission (extended PWN) seen up to at least2.4' south of the pulsar, with a softer spectrum (Gamma_ext=1.9), and the 0.5-8keV luminosity L_ext~10^{33}-10^{34} ergs s^{-1}. We also measured the pulsar'sproper motion using archival VLA data: \mu_\alpha=23.0+/-2.5 mas yr^{-1},  \mu_\delta=-3.9+/-3.3 mas yr^{-1}, which corresponds to the transversevelocity v_perp=440 km s^{-1}. The direction of the proper motion isapproximately parallel to the elongation of the compact PWN, but it is nearlyperpendicular to that of the extended PWN and to the direction towards thecenter of the bright VHE gamma-ray source HESS J1825-137, which is likelypowered by PSR B1823-13.

Chandra Observation of PSR B1823-13 and its Pulsar Wind Nebula