The structure and kinematics of the ionized gas within the starburst nucleus of M82 are investigated through high spectral (30 km s(-1)) and spatial (2'') resolution observations of the 12.8 mu m line of [Ne II]. The structure is dominated by a nuclear ring and two ionized ''lanes.'' These components are found in other barred galaxies, supporting the suggestion that M82 may contain a nuclear bar. The ionized ring is approximately circular with a radius of 86 pc and an inclination of 73 degrees. The center of the ring closely corresponds to M82's kinematic center and is located at 9(h)51(m)43(5).4 +/-0''.5, +69 degrees 55'00''.1 +/-0''.5 (1950). M82's systemic velocity is determined to be 200 +/- 7 km s(-1). The rotation curve becomes flat, with a terminal velocity of 112 +/- 5 km s(-1) beyond 5'' from the center. There are signs of outflow along the minor axis with a velocity similar to 55 km s(-1). M82's rotation curve is searched for signs of the x(1) and x(2) orbital families which make up a nuclear bar. Unfortunately, the orientation of M82's proposed bar is such that the expected rotation curves from barred or circular orbits are very similar. Hints of x(1) and x(2) orbits are found in M82, but kinematic identification of a bar remains elusive. Observations of [Ar III] (8.99 mu m), [S IV] (10.51 mu m), and hydrogen Br alpha measure a lower level of excitation than that found in most Galactic H II regions, and which remains fairly constant across the galaxy. The total neon abundance is 1-2 times solar. The temperature of the stars ionizing the starburst regions is approximately 33,000 K. The Br alpha flux requires about 6 x 10(53) Lyman continuum photons per second. Star formation has occurred either during a very brief burst (or bursts) or with an initial mass function with an upper mass cutoff approximately 25 M..
