We report the discovery of PSR J2022+3842, a 24 ms radio and X-ray pulsar in the supernova remnant G76.9+1.0, in observations with the Chandra X-ray telescope, the Robert C. Byrd Green Bank Radio Telescope, and the Rossi X-ray Timing Explorer (RXTE). The pulsar's spin-down rate implies a rotation-powered luminosity Edot = 1.2 x 10^{38} erg/s, a surface dipole magnetic field strength B_s = 1.0 x 10^{12} G, and a characteristic age of 8.9 kyr. PSR J2022+3842 is thus the second-most energetic Galactic pulsar known, after the Crab, as well as the most rapidly-rotating young, radio-bright pulsar known. The radio pulsations are highly dispersed and broadened by interstellar scattering, and we find that a large (delta-f / f ~= 1.9 x 10^{-6}) spin glitch must have occurred between our discovery and confirmation observations. The X-ray pulses are narrow (0.06 cycles FWHM) and visible up to 20 keV, consistent with magnetospheric emission from a rotation-powered pulsar. The Chandra X-ray image identifies the pulsar with a hard, unresolved source at the midpoint of the double-lobed radio morphology of SNR G76.9+1.0 and embedded within faint, compact X-ray nebulosity. The spatial relationship of the X-ray and radio emissions is remarkably similar to extended structure seen around the Vela pulsar. The combined Chandra and RXTE pulsar spectrum is well-fitted by an absorbed power-law model with column density N_H = (1.7\pm0.3) x 10^{22} cm^{-2} and photon index Gamma = 1.0\pm0.2; it implies that the Chandra point-source flux is virtually 100% pulsed. For a distance of 10 kpc, the X-ray luminosity of PSR J2022+3842 is L_X(2-10 keV) = 7.0 x 10^{33} erg s^{-1}. Despite being extraordinarily energetic, PSR J2022+3842 lacks a bright X-ray wind nebula and has an unusually low conversion efficiency of spin-down power to X-ray luminosity, L_X/Edot = 5.9 x 10^{-5}.