Based on more than four weeks of continuous high cadence photometric monitoring of several hundred members of the young cluster NGC 2264 with two space telescopes, NASA's Spitzer and the CNES CoRoT (Convection, Rotation, and planetary Transits), we provide high quality, multi-wavelength light curves for young stellar objects (YSOs) whose optical variability is dominated by short duration flux bursts, which we infer are due to enhanced mass accretion rates. These light curves show many brief -- several hour to one day -- brightenings at optical and near-infrared (IR) wavelengths with amplitudes generally in the range 5-50% of the quiescent value. Typically, a dozen or more of these bursts occur in a thirty day period. We demonstrate that stars exhibiting this type of variability have large ultraviolet (UV) excesses and dominate the portion of the u-g vs. g-r color-color diagram with the largest UV excesses. These stars also have large Halpha equivalent widths, and either centrally peaked, lumpy Halpha emission profiles or profiles with blue-shifted absorption dips associated with disk or stellar winds. Light curves of this type have been predicted for stars whose accretion is dominated by Rayleigh-Taylor instabilities at the boundary between their magnetosphere and inner circumstellar disk, or where magneto-rotational instabilities modulate the accretion rate from the inner disk. Amongst the stars with the largest UV excesses or largest Halpha equivalent widths, light curves with this type of variability greatly outnumber light curves with relatively smooth sinusoidal variations associated with long-lived hot spots. We provide quantitative statistics for the average duration and strength of the accretion bursts and for the fraction of the accretion luminosity associated with these bursts. ; Comment: Accepted for publication in AJ. 39 pages; 6 tables; 25 figures, many of which are highly degraded to meet size limits. Please download the regular resolution version at http://web.ipac.caltech.edu/staff/amc/staufferetal2014.pdf