We perform a systematic search for long-term extreme variability quasars (EVQs) in the overlapping Sloan Digital Sky Survey and 3 Year Dark Energy Survey imaging, which provide light curves spanning more than 15 years. We identified similar to 1000 EVQs with a maximum change in g-band magnitude of more than 1 mag over this period, about 10% of all quasars searched. The EVQs have L-bol similar to 10(45)-10(47) erg s(-1) and L/L-Edd similar to 0.01-1. Accounting for selection effects, we estimate an intrinsic EVQ fraction of similar to 30%-50% among all g less than or similar to 22 quasars over a baseline of similar to 15 yr. We performed detailed multi-wavelength, spectral, and variability analyses for the EVQs and compared them to their parent quasar sample. We found that EVQs are distinct from a control sample of quasars matched in redshift and optical luminosity: (1) their UV broad emission lines have larger equivalent widths; (2) their Eddington ratios are systematically lower; and (3) they are more variable on all timescales. The intrinsic difference in quasar properties for EVQs suggests that internal processes associated with accretion are the main driver for the observed extreme long-term variability. However, despite their different properties, EVQs seem to be in the tail of a continuous distribution of quasar properties, rather than standing out as a distinct population. We speculate that EVQs are normal quasars accreting at relatively low rates, where the accretion flow is more likely to experience instabilities that drive the changes in flux by a factor of a few on multi-year timescales.