We present J, H, CH_4 short (1.578 micron), CH_4 long (1.652 micron) and K_s-band images of the dust ring around the 10 Myr old star HR 4796 A obtained using the Near Infrared Coronagraphic Imager (NICI) on the Gemini-South 8.1 meter Telescope. Our images clearly show for the first time the position of the star relative to its circumstellar ring thanks to NICI's translucent focal plane occulting mask. We employ a Bayesian Markov Chain Monte Carlo method to constrain the offset vector between the two. The resulting probability distribution shows that the ring center is offset from the star by 16.7+/-1.3 milliarcseconds along a position angle of 26+/-3 degrees, along the PA of the ring, 26.47+/-0.04 degrees. We find that the size of this offset is not large enough to explain the brightness asymmetry of the ring. The ring is measured to have mostly red reflectivity across the JHK_s filters, which seems to indicate micron-sized grains. Just like Neptune's 3:2 and 2:1 mean-motion resonances delineate the inner and outer edges of the classical Kuiper Belt, we find that the radial extent of the HR 4796 A and Fomalhaut rings could correspond to the 3:2 and 2:1 mean-motion resonances of hypothetical planets at 54.7 AU and 97.7 AU in the two systems, respectively. A planet orbiting HR 4796 A at 54.7 AU would have to be less massive than 1.6 Mjup so as not to widen the ring too much by stirring. ; Comment: Accepted to A&A for publication on April 23, 2014 (15 pages, 9 figures, 4 tables)