We present the first Laser Guide Star Adaptive Optics (LGS-AO) observations of the Galactic center. LGS-AO has dramatically improved the quality and robustness with which high angular resolution infrared images of the Galactic center can be obtained with the W. M. Keck II 10-meter telescope. Specifically, Strehl ratios of 0.7 and 0.3 at L'[3.8 micron ] and K'[2.1 micron ], respectively, are achieved in these LGS-AO images; these are at least a factor of two higher and a factor of four to five more stable against atmospheric fluctuations than the Strehl ratios delivered thus far with the Keck Natural Guide Star AO system on the Galactic center. Furthermore, these observations are the first that cover a large area (76'' * 76'') surrounding the central black hole at diffraction-limited resolution for an 8-10 meter class telescope. During our observations, the infrared counterpart to the central supermassive black hole, Sgr A*-IR, showed significant infrared intensity variations, with observed L' magnitudes ranging from 12.6 to 14.5 mag and a decrease in flux density of a factor of two over an 8 minute interval. The faintest end of our L' detections, 1.3 mJy (dereddened), is the lowest level of emission yet observed for this source by a factor of 3. No significant variation in the location of SgrA*-IR is detected as a function of either wavelength or intensity. Previous claims of such positional variations are easily attributable to a nearby (0.'' 09 or 720 AU, projected), extended, very red source, which we suggest arises from a locally heated dust feature. Near a peak in its intensity, we obtained the first measurement of SgrA*-IR's K'-L' color; its K'-L' of 3.0 +/- 0.2 mag (observed) or 1.4 +/- 0.2 (dereddened) corresponds to an intrinsic spectral index of alpha -0.5 +/- 0.3 for F_ nu nu alpha . This is significantly bluer than other recent infrared measurements from the literature, which suggest alpha = -4 +/- 1. Because our measurement was taken at a time when Sgr A* was 6 times brighter in the infrared than the other measurements, we posit that the spectral index of the emission arising from the vicinity of our Galaxy's central black hole may depend on the strength of the flare, with stronger flares giving rise to a higher fraction of high energy electrons in the emitting region.