Micro-X is a proposed sounding rocket experiment that will combine a transition-edge-sensor X-ray-microcalorimeter array with a conical imaging mirror to obtain high-spectral-resolution images of extended and point X-ray sources. We describe the payload and the science targeted by this mission including the discussion of three possible Micro-X targets: the Puppis A supernova remnant, the Virgo Cluster, and Circinus X-1. For example, a Micro-X observation of the bright eastern knot of Puppis A will obtain a line-dominated spectrum with 90,000 counts collected in 300 seconds at 2 eV resolution across the 0.3-2.5 keV band. Micro-X will utilize plama diagnostics to determine the thermodynamic and ionization state of the plasma, to search for line shifts and broadening associated with dynamical processes, and seek evidence of ejecta enhancement. For clusters of galaxies, Micro-X can uniquely study turbulence and the temperature distribution function. For binaries, Micro-X's high resolution spectra will separate the different processes contributing to the Fe K lines at 6 keV and give a clear view of the geometry of the gas flows and circumstellar gas.