We present an H-band image of the light scattered from circumstellar dust around the nearby (10 pc) young M star AU Microscopii (AU Mic, GJ 803, HD 197481), obtained with the Keck adaptive optics system. We resolve the disk both vertically and radially, tracing it over 17-60 AU from the star. Our high angular resolution (0.04 or 0.4 AU per resolution element) observations thus offer the opportunity to probe the morphology of the AU Mic debris disk on solar system scales. We identify substructures (dust clumps and gaps) in the disk that may point to the existence of orbiting planets. No planets are seen in our H-band image down to a limiting mass of 1M_J at >20 AU, although the existence of smaller planets cannot be excluded from the current data. Modeling of the disk surface brightness distribution at H band and R band, and of the optical to submillimeter spectral energy distribution, allows us to constrain the disk geometry and the dust grain properties. We confirm the nearly edge-on orientation of the disk inferred from previous observations and deduce that the disk may be clear inward of 1-10 AU. We find evidence for a lack of small grains at <50 AU, either as a result of grain growth or because of destruction by Poynting-Robertson and/or corpuscular drag. A change in the power-law index of the surface brightness profile is observed near 33 AU, similar to a feature known in the profile of the β Pic circumstellar debris disk. By comparing the timescales for interparticle collisions and Poynting-Robertson drag between the two systems, we argue that the breaks are linked to one or both of these processes.