The Large and Small Magellanic Clouds (LMC and SMC) are unique local laboratories for studying the formation and evolution of small galaxies in exquisite detail. The Survey of the MAgellanic Stellar History (SMASH) is an NOAO community DECam survey of the Clouds mapping 480 square degrees (distributed over ~2400 square degrees at ~20% filling factor) to ~24th mag in ugriz with the goal of identifying broadly distributed, low surface brightness stellar populations associated with the stellar halos and tidal debris of the Clouds. SMASH will also derive spatially-resolved star formation histories covering all ages out to large radii from the MCs that will further complement our understanding of their formation. Here, we present a summary of the survey, its data reduction, and a description of the first public Data Release (DR1). The SMASH DECam data have been reduced with a combination of the NOAO Community Pipeline, PHOTRED, an automated PSF photometry pipeline based mainly on the DAOPHOT suite, and custom calibration software. The attained astrometric precision is ~15 mas and the accuracy is ~2 mas with respect to the Gaia DR1 astrometric reference frame. The photometric precision is ~0.5-0.7% in griz and ~1% in u with a calibration accuracy of ~1.3% in all bands. The median 5 sigma point source depths in ugriz bands are 23.9, 24.8, 24.5, 24.2, 23.5 mag. The SMASH data already have been used to discover the Hydra II Milky Way satellite, the SMASH 1 old globular cluster likely associated with the LMC, and very extended stellar populations around the LMC out to R~18.4 kpc. SMASH DR1 contains measurements of ~100 million objects distributed in 61 fields. A prototype version of the NOAO Data Lab provides data access, including a data discovery tool, SMASH database access, an image cutout service, and a Jupyter notebook server with example notebooks for exploratory analysis.