We present a detailed investigation of the variability of 428 C IV and 235 Si IV broad absorption line (BAL) troughs identified in multi-epoch observations of 291 quasars by the Sloan Digital Sky Survey-I/II/III. These observations primarily sample rest-frame timescales of 1-3.7 yr over which significant rearrangement of the BAL wind is expected. We derive a number of observational results on, e.g., the frequency of BAL variability, the velocity range over which BAL variability occurs, the primary observed form of BAL-trough variability, the dependence of BAL variability upon timescale, the frequency of BAL strengthening versus weakening, correlations between BAL variability and BAL-trough profiles, relations between C IV and Si IV BAL variability, coordinated multi-trough variability, and BAL variations as a function of quasar properties. We assess implications of these observational results for quasar winds. Our results support models where most BAL absorption is formed within an order-of-magnitude of the wind-launching radius, although a significant minority of BAL troughs may arise on larger scales. We estimate an average lifetime for a BAL trough along our line-of-sight of a few thousand years. BAL disappearance and emergence events appear to be extremes of general BAL variability, rather than being qualitatively distinct phenomena. We derive the parameters of a random-walk model for BAL EW variability, finding that this model can acceptably describe some key aspects of EW variability. The coordinated trough variability of BAL quasars with multiple troughs suggests that changes in "shielding gas" may play a significant role in driving general BAL variability.