We analyze the statistical properties of the circumnuclear H II regions of a sample of 52 nearby galaxies (v<1000 km s-1) from archival HST/NICMOS H-band and Paα (1.87 μm) observations at unprecedented spatial resolutions of between 1 and 30 pc. We catalog H II regions from the continuum-subtracted Paα images and find H II regions in the central regions of most galaxies, and more than a hundred in each of eight galaxies. In contrast to disk H II regions, the physical properties (luminosity and size) of individual circumnuclear H II regions do not vary strongly with the morphological type of the host galaxy, nor does the number of circumnuclear H II regions per unit area. The Hα luminosity within the central kiloparsec, as derived from H II region emission, is significantly enhanced in early-type (S0/a-Sb) galaxies. We find evidence that bars increase the circumnuclear star formation, presumably by funneling gas from the disk toward the nucleus. Barred galaxies exhibit enhanced luminosities of the brightest H II region, the central kiloparsec Hα luminosities (an effect mostly due to the early-type galaxies in our sample), and the star formation rates per unit stellar mass (which could also be understood as the integral equivalent widths of Paα) over the central kiloparsec with respect to nonbarred galaxies. We fit the luminosity functions (LFs) and diameter distributions of the circumnuclear H II regions in eight galaxies where we can catalog enough H II regions to do so in a meaningful way. We use power laws and find that the fitted slopes of the H II region LF are exactly in the previously found ranges and even confirm a trend with steeper slopes in galaxies of earlier morphological type. This implies that the physical processes giving rise to enhanced star formation in the circumnuclear regions of galaxies must be similar to those in disks. Based on observations with the NASA/ESA Hubble Space Telescope, obtained from the data archive at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555.