We present new near-infrared observations of the starburst galaxy M82, including high-resolution J, H, and K imagery, 1-5 mum spectrophotometry and photometry, and 1.05-1.35 mum spectroscopy. These measurements are used to examine the stellar population in the galaxy and conditions in the interstellar medium. We combine our observations with results from the literature to derive a new set of constraints for starburst models, which are presented in a companion paper. Our measurements of Paschen line fluxes in combination with the strengths of other recombination lines allow study of the reddening in M82. We find that a uniform foreground screen model for the dust geometry cannot explain the relative line strengths. A variety of other models can fit the data reasonably well, such as a nonuniform foreground screen or a homogeneous mix of dust and gas. The derived properties of the galaxy are not strongly dependent on which of these models is selected; we derive an ionizing flux of almost-equal-to 10(54) s-1 and an absolute magnitude of the starburst region of M(K) < -22.5. Depending on the reddening model, the visual extinction to the nucleus lies between A(V) = 12 and A(V) = 27; we show that the nucleus does not coincide with any feature that is bright at visual wavelengths. The stellar H2O index in the nuclear region is less than 0.06 and the CO index is almost-equal-to 0.21; contamination of the CO band by emission by hot dust must be small. We use several lines in the 1.3 mum region to derive a lower limit to the electron density of n(e) > 10(4.5) cm-3 in the [Fe II] emission region. The [Fe II] line fluxes are consistent with excitation by fast shocks incident on dense gas. We report the tentative detection of several lines of H-2, which, if confirmed, implicate UV fluorescence as the excitation mechanism.