The Solar Dynamics Observatory Extreme Ultraviolet Variability Experiment (SDO/EVE) provides “Sun-as-a-star” data corresponding to the variability of the Sun’s irradiance in the XUV and EUV wavelengths (from 0.1 to 106 nm). Using EVE light curves in the 304 Angstrom line, we study 2049 solar flares from 2010 April 30 to 2014 May 26. We present an algorithm for fitting the flare light curves in the 304 Angstrom line, emitted by He II at around 50000 K from the chromosphere and transition region and therefore representative of the dominant source of radiation in a solar flare. We use this algorithm to identify particularly high signal-to-noise flare light curves within the database, with representatives from C, M, and X flare classes. The parameters of the model associated with each flare can be used to identify features such as secondary peaks in the decay phase. In addition, we devise a morphological classification scheme based on flare “impulsiveness” and apply the scheme to a sub-sample of the flares. While a similar method has been used in the past to classify stellar flares, it has yet to be extensively applied to solar flare light curves. Morphological variations in flare development and their relationship to impulsiveness are studied using SDO Atmospheric Imaging Assembly (AIA) instrument data in the 1600 Angstrom line. A comparison is made to several solar flare properties including duration, peak X-ray flux, and quasi-periodic pulsation (QPP) period, among others. Using the modeling algorithm and impulsiveness classification scheme in the chromospheric 304 Angstrom line, it may be possible to identify and study solar and stellar flare features not revealed by traditional methods, particularly when Sun-as-a-star light curves are studied together with full-disk images.