We report 1''.8 resolution imaging of 2.17 mum H I Brgamma and 2.12 mum H2 1-0 S(1) line emission and 2.15 mum continuum emission toward the nuclear region of the ''hot spot'' spiral galaxy NGC 1808. The data, taken with a high-resolution imaging spectrometer, show that the Brgamma emission originates from several distinct emission knots which are distributed throughout the circumnuclear region extending approximately 14'' (750 pc) along the major axis of the galaxy. Most of the H2 emission is confined to a region within approximately 4'' (200 pc) of the nucleus. The 2.15 mum continuum emission is distributed smoothly (at 1''.8 resolution) throughout the circumstellar region, is not produced in discrete hot spots, and peaks sharply at the galactic nucleus. The Brgamma emission knots are spatially well correlated with the set of compact radio sources that are present in the circumnuclear region. Our Brgamma data imply that between 10% and 50% of the compact circumnuclear 5 GHz radio emission is thermal, free-free emission. The Brgamma emission knots do not coincide with the locations of optical ''hot-spots'' observed in NGC 1808. From a comparison of the Brgamma and Halpha emission knots we conclude that the Brgamma knots trace the actual sites of star-formation activity, while the optical hot spots are mainly directions of low foreground extinction. We use our high-resolution near-infrared observations together with radio and far-infrared continuum observations and theoretical models of evolving star clusters to constrain the properties of the distinct star-formation complexes in NGC 1808. The data suggest that star-formation has been proceeding, at rates of approximately 1 M. yr-1, for approximately-greater-than 5 x 10(7) yr in the nuclear cluster, and for approximately-lesser-than 10(7) yr in the circumnuclear star clusters. The data are consistent with stellar initial mass functions of the form M(-alpha) with alpha approximately 2.5 which are truncated at upper mass cutoffs of approximately 30 M.. The IMFs may extend to small (approximately 0.1 M.) lower mass limits if the star-formation activity in the star-forming complexes is sufficiently efficient. Much of the extranuclear 2.15 mum continuum emission is probably produced by evolved bulge stars which are much older than the lifetimes of the circumnuclear star-forming clusters. However, these clusters also contribute significantly to the observed 2.15 mum continuum. The intense 2.15 mum continuum emission at the nucleus is very likely dominated by red giants and supergiants recently produced in the nuclear star-forming cluster. We argue that most of the bolometric luminosity generated in the center of NGC 1808 is produced by the star-forming clusters. There is no near-infrared evidence for an embedded active galactic nucleus in NGC 1808.