The Apache Point Observatory Galactic Evolution Experiment (APOGEE) has built the largest moderately high-resolution (R ≈ 22,500) spectroscopic map of the stars across the Milky Way, and including dust-obscured areas. The APOGEE Stellar Parameter and Chemical Abundances Pipeline (ASPCAP) is the software developed for the automated analysis of these spectra. ASPCAP determines atmospheric parameters and chemical abundances from observed spectra by comparing observed spectra to libraries of theoretical spectra, using χ2 minimization in a multidimensional parameter space. The package consists of a fortran90 code that does the actual minimization and a wrapper IDL code for book-keeping and data handling. This paper explains in detail the ASPCAP components and functionality, and presents results from a number of tests designed to check its performance. ASPCAP provides stellar effective temperatures, surface gravities, and metallicities precise to 2%, 0.1 dex, and 0.05 dex, respectively, for most APOGEE stars, which are predominantly giants. It also provides abundances for up to 15 chemical elements with various levels of precision, typically under 0.1 dex. The final data release (DR12) of the Sloan Digital Sky Survey III contains an APOGEE database of more than 150,000 stars. ASPCAP development continues in the SDSS-IV APOGEE-2 survey.