Mapping Nearby Galaxies at Apache Point Observatory (MaNGA) is acquiring integral-field spectroscopy for the largest sample of galaxies to date. By 2020, the MaNGA Survey --- one of three core programs in the fourth-generation Sloan Digital Sky Survey (SDSS-IV) --- will have observed a statistically representative sample of 10$^4$ galaxies in the local Universe ($z\lesssim0.15$). In addition to a robust data-reduction pipeline (DRP), MaNGA has developed a data-analysis pipeline (DAP) and software package yielding higher-level data products. This paper describes the first public release of the DAP software and assesses its output provided by the recent SDSS Data Release 15 (DR15). The DAP has focused on measurements that are as close as possible to the data and require minimal model-based interpretations or assumptions for reproduction. For DR15, these measurements include stellar kinematics (velocity and velocity dispersion), emission-line properties (kinematics, fluxes, and equivalent widths), and spectral indices (e.g., D4000 and the Lick indices). We provide an overview of how these measurements are made in the MaNGA DAP, including discussion of its software design, workflow, performance, and output data model. The MaNGA DAP provides robust measurements and errors for the vast majority (>99%) of analyzed spectra. We summarize assessments of the precision and accuracy of our measurements as a function of signal-to-noise and other salient metrics, noting that additional analysis and discussion of emission-line diagnostics are provided in our companion paper, Belfiore et al. The MaNGA DAP software is publicly available and we encourage community input on and contributions to its development.