Star formation is one of the least understood processes in cosmic evolution. It is difficult to formulate a general theory for star formation in part because of the wide range of physical processes involved. The interstellar gas out of which stars form is a supersonically turbulent plasma governed by magnetohydrodynamics. This is hard enough by itself, since we do not understand even subsonic hydrodynamic turbulence very well, let alone supersonic non-ideal MHD turbulence. However, the behavior of star-forming clouds in the ISM is also obviously influenced by gravity, which adds complexity, and by both continuum and line radiative processes. Finally, the behavior of star-forming clouds is influenced by a wide variety of chemical processes, including formation and destruction of molecules and dust grains (which changes the thermodynamic behavior of the gas) and changes in ionization state (which alter how strongly the gas couples to magnetic fields). As a result of these complexities, there is nothing like a generally agreed-upon theory of star formation, as there is for stellar structure. Instead, we are forced to take a much more phenomenological approach. These notes provide an introduction to our current thinking about how star formation works. ; Comment: To appear in the XVth Special Courses of the National Observatory of Rio de Janeiro, 49 pages, 11 figures, AIP conference format. This is a pedagogic introduction to star formation science, intended for beginning grad students or advanced undergraduates