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We present the selection and classification of over a thousand ultraviolet (UV) variable sources discovered in ∼40 deg2 of GALEX Time Domain Survey (TDS) NUV images observed with a cadence of 2 days and a baseline of observations of ∼3 years. The GALEX TDS fields were designed to be in spatial and temporal coordination with the Pan-STARRS1 Medium Deep Survey, which provides deep optical imaging and simultaneous optical transient detections via image differencing. We characterize the GALEX photometric errors empirically as a function of mean magnitude, and select sources that vary at the 5σ level in at least one epoch. We measure the statistical properties of the UV variability, including the structure function on timescales of days and years. We report classifications for the GALEX TDS sample using a combination of optical host colors and morphology, UV light curve characteristics, and matches to archival X-ray, and spectroscopy catalogs. We classify 62% of the sources as active galaxies (358 quasars and 305 active galactic nuclei), and 10% as variable stars (including 37 RR Lyrae, 53 M dwarf flare stars, and 2 cataclysmic variables). We detect a large-amplitude tail in the UV variability distribution for M-dwarf flare stars and RR Lyrae, reaching up to ∣Δm∣ = 4.6 mag and 2.9 mag, respectively. The mean amplitude of the structure function for quasars on year timescales is five times larger than observed at optical wavelengths. The remaining unclassified sources include UV-bright extragalactic transients, two of which have been spectroscopically confirmed to be a young core-collapse supernova and a flare from the tidal disruption of a star by dormant supermassive black hole. We calculate a surface density for variable sources in the UV with NUV < 23 mag and ∣Δm∣ > 0.2 mag of ∼8.0, 7.7, and 1.8 deg-2 for quasars, active galactic nuclei, and RR Lyrae stars, respectively. We also calculate a surface density rate in the UV for transient sources, using the effective survey time at the cadence appropriate to each class, of ∼15 and 52 deg-2 yr-1 for M dwarfs and extragalactic transients, respectively. [ABSTRACT FROM AUTHOR]