The structure in cells responsible for modifying proteins after synthesis is the Golgi apparatus. This organelle processes, sorts, and packages proteins produced in the endoplasmic reticulum, often adding carbohydrate groups in a process called glycosylation. After modifications, the Golgi apparatus sends these proteins to their final destinations, either inside or outside the cell, through vesicles. This trafficking is vital for proper cellular function and ensures that proteins are correctly folded and functional, playing a crucial role in cellular communication and metabolism.