We present ugriz light curves for 146 spectroscopically-confirmed or spectroscopically-probable Type Ia supernovae (SNe) from the 2005 season of the Sloan Digital Sky Survey-II Supernova (SN) survey. The light curves have been constructed using a photometric technique that we call scene modeling, which is described in detail here; the major feature is that SN brightnesses are extracted from a stack of images without spatial resampling or convolution of the image data. This procedure produces accurate photometry along with accurate estimates of the statistical uncertainty, and can be used to derive photometry taken with multiple telescopes. We discuss various tests of this technique that demonstrate its capabilities. We also describe the methodology used for the calibration of the photometry, and present calibrated magnitudes and fluxes for all of the spectroscopic SNe Ia from the 2005 season.