Andromeda X (And X) is a newly discovered low-luminosity M31 dwarf spheroidal galaxy (dSph) found by Zucker et al. in the Sloan Digital Sky Survey (SDSS; York et al.). In this paper, we present the first spectroscopic study of individual red giant branch stars in And X, as a part of the Spectroscopic and Photometric Landscape of Andromeda's Stellar Halo (SPLASH) Survey. Using the Keck II telescope and multiobject DEIMOS spectrograph, we target two spectroscopic masks over the face of the galaxy and measure radial velocities for ~100 stars with a median accuracy of σ v ~ 3 km s-1. The velocity histogram for this field confirms three populations of stars along the sight line: foreground Milky Way dwarfs at small negative velocities, M31 halo red giants over a broad range of velocities, and a very cold velocity "spike" consisting of 22 stars belonging to And X with v rad = -163.8 ± 1.2 km s-1. By carefully considering both the random and systematic velocity errors of these stars (e.g., through duplicate star measurements), we derive an intrinsic velocity dispersion of just σ v = 3.9 ± 1.2 km s-1 for And X, which for its size, implies a minimum mass-to-light ratio of M/LV = 37+26 -19 assuming that the mass traces the light. Based on the clean sample of member stars, we measure the median metallicity of And X to be [Fe/H] = -1.93 ± 0.11, with a slight radial metallicity gradient. The dispersion in metallicity is large, σ([Fe/H]phot) = 0.48, possibly hinting that the galaxy retained much of its chemical enrichment products. And X has a total integrated luminosity (MV = -8.1 ± 0.5) that straddles the classical Local Group dSphs and the new SDSS ultra-low luminosity galaxies. The galaxy is among the most metal-poor dSphs known, especially relative to those with MV < -8, and has the second lowest intrinsic velocity dispersion of the entire sample. Our results suggest that And X is less massive by a factor of 4 when compared to Milky Way dSphs of comparable luminosity (e.g., Draco and Ursa Minor). We discuss the potential for better understanding the formation and evolution mechanisms for M31's system of dSphs through (current) kinematic and chemical abundance studies, especially in relation to the Milky Way sample. Data presented herein were obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California, and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W. M. Keck Foundation. Based on observations made with the Nordic Optical Telescope, operated on the island of La Palma jointly by Denmark, Finland, Iceland, Norway, and Sweden, in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias; these observations were funded by the Optical Infrared Coordination Network (OPTICON), a major international collaboration supported by the Research Infrastructures Programme of the European Commission's Sixth Framework Programme.