We present the Spectroscopic Imaging survey in the near-infrared (near-IR) with SINFONI (SINS) of high-redshift galaxies. With 80 objects observed and 63 detected in at least one rest-frame optical nebular emission line, mainly H Delta *a, SINS represents the largest survey of spatially resolved gas kinematics, morphologies, and physical properties of star-forming galaxies at z ~ 1-3. We describe the selection of the targets, the observations, and the data reduction. We then focus on the 'SINS H Delta *a sample,' consisting of 62 rest-UV/optically selected sources at 1.3 < z < 2.6 for which we targeted primarily the H Delta *a and N II emission lines. Only 30% of this sample had previous near-IR spectroscopic observations. The galaxies were drawn from various imaging surveys with different photometric criteria; as a whole, the SINS H Delta *a sample covers a reasonable representation of massive M 1010 M star-forming galaxies at z 1.5-2.5, with some bias toward bluer systems compared to pure K-selected samples due to the requirement of secure optical redshift. The sample spans 2 orders of magnitude in stellar mass and in absolute and specific star formation rates, with median values 3 X 1010 M, 70 M yr-1, and 3 Gyr-1. The ionized gas distribution and kinematics are spatially resolved on scales ranging from 1.5 kpc for adaptive optics assisted observations to typically 4-5 kpc for seeing-limited data. The H Delta *a morphologies tend to be irregular and/or clumpy. About one-third of the SINS H Delta *a sample galaxies are rotation-dominated yet turbulent disks, another one-third comprises compact and velocity dispersion-dominated objects, and the remaining galaxies are clear interacting/merging systems; the fraction of rotation-dominated systems increases among the more massive part of the sample. The H Delta *a luminosities and equivalent widths suggest on average roughly twice higher dust attenuation toward the H II regions relative to the bulk of the stars, and comparable current and past-averaged star formation rates.