We present results on the Soft Gamma Repeater (SGR) 0501+4516, discovered by the SWIFT Burst Alert Telescope (BAT) on 2008 August 22. More than 50 bursts were identified from this source, out of which 18 bursts had enough counts to carry out spectral analysis. We performed time-averaged spectral analysis on these 18 bursts using 8 models, among which the cut-off powerlaw and the two-blackbody models provided the best fit in the 15-150 keV energy range. The cut-off powerlaw model fit yields a mean photon index Gamma_{CPL} = 0.54+/-0.11 and a cut-off energy E_C = 19.1+/-1.8 keV for the bursts. The mean hard and soft blackbody temperatures are found to be kT_{BB_h} = 12.8+/-0.7 keV and kT_{BB_s} = 4.6+/-0.5 keV, respectively, and are anti-correlated with the square of the radii of the hard and soft emitting regions (R_{BB_h} and R_{BB_s}) as R_{BB_h}^2 $∝$ kT^{-5.8} and R_{BB_s}^2 $∝$ kT^{-2.7}, respectively. The soft and hard component temperatures with different indices support the idea of two distinct emitting regions with the hard component corresponding to a smaller radius and the soft component corresponding to a larger radius, which further corroborate the idea of the propagation of extraordinary (E) and ordinary (O) mode photons across the photosphere, as predicted in the magnetar model. We notice strong burst fluence-duration correlation as well as hardness ratio-duration and hardness ratio-fluence anti-correlations for the SGR 0501+4516 bursts. The burst fluences range from ~ 4.4\times10^{-9} ergs/cm^-2 to ~ 2.7\times10^{-6} ergs/cm^{-2}, consistent with those observed for typical short SGR bursts. Comment: 13 pages, 8 figures, 4 tables. Accepted for publication in the Astrophysical Journal