The nuclear star cluster of the Galaxy is an important template for understanding its extragalactic counterparts, which can currently not be resolved into individual stars. Important drawbacks of observations of the Galactic center are, however, the presence of strong and spatially highly variable interstellar extinction and extreme crowding of the sources, which makes the use of adaptive optics techniques necessary. Both points pose serious obstacles to precise photometry that is needed for analyzing the stellar population. The aims of this work are to provide accurate photometry in multiple near-infrared broadband filters, to determine the power-law index of the extinction-law toward the central parsec of the Galaxy, to provide measurements of the absolute extinction toward the Galactic center, and finally to measure the spatial variability of extinction on arcsecond scales. We use observations of the central parsec of the Milky Way that were obtained with the near-infrared camera and adaptive optics system NAOS/CONICA at the ESO VLT unit telescope 4. The photometric method takes into account anisoplanatic effects and limits the corresponding systematic uncertainties to <=2%. Absolute values for the extinction in the H, Ks, and L'-bands as well as of the power-law indices of the H to Ks and Ks to L' extinction-laws are measured based on the well-known properties of red clump stars. Extinction maps are derived based on H-Ks and Ks-L' colors. We present Ks-band photometry for 7700 stars, and additionally photometry for stars detected in the H and/or L'-bands. From a number of recently published values we compute a mean distance of the Galactic center of R_0=8.03+/-0.15 kpc, which has an uncertainty of just 2%. Based on this R_0 and on the RC method, we derive absolute mean extinction values toward the central parsec of the Galaxy of A_H=4.48+/-0.13 mag, A_Ks=2.54+/-0.12 mag, and A_L'=1.27+/-0.18 mag. We estimate values of the power-law indices of the extinction-law of alpha _H-Ks=2.21+/-0.24 and alpha _Ks-L'=1.34+/-0.29. A Ks-band extinction map for the Galactic center is computed based on this extinction law and on stellar H-Ks colors. Both its statistical and systematic uncertainties are estimated to be <10%. Extinction in this map derived from stellar color excesses is found to vary on arcsecond scales, with a mean value of A_Ks=2.74+/-0.30 mag. Mean extinction values in a circular region with 0.5" radius centered on Sagittarius A* are A_H, SgrA*=4.35+/-0.12, A_Ks, SgrA*=2.46+/-0.03, and A_L', SgrA*=1.23+/-0.08.