We present results from 1351 high-resolution spectra of 1215 stars in the Orion Nebula Cluster (ONC) and the surrounding Orion 1c association, obtained with the Hectochelle multiobject echelle spectrograph on the 6.5 m MMT. We confirmed 1111 stars as members, based on their radial velocity and/or H alpha emission. The radial velocity distribution of members shows a dispersion of image km s super(-1). We found a substantial north-south velocity gradient and spatially coherent structure in the radial velocity distribution, similar to that seen in the molecular gas in the region. We also identified several binary and high velocity stars, a region exhibiting signs of triggered star formation, and a possible foreground population of stars somewhat older than the ONC. Stars without infrared excesses (as detected with the IRAC instrument on the Spitzer Space Telescope) exhibit a wider spread in radial velocity than the infrared excess stars; this spread is mostly due to a blueshifted population of stars that may constitute a foreground population. We also identify some accreting stars, based on H alpha , that do not have detectable infrared excesses with IRAC, and thus are potential transitional disk systems (objects with inner disk holes). We propose that the substructure seen in both the stellar and gaseous components is the result of nonuniform gravitational collapse to a filamentary distribution of gas. The spatial and kinematic correlation between the stellar and gaseous components suggests that the region is very young, probably only image1 crossing time old or less, to avoid shock dissipation and gravitational interactions which would tend to destroy the correlation between stars and gas.