Whose main job is to break down worn-out cell parts and other waste in an animal cell?

The correct answer is lysosome.
A lysosome is a membrane-bound organelle with many different functions:
·     degradation- lysosome contains hydrolytic enzymes that can break down (hydrolysis at low pH) many kinds of biomolecules.
·     secretion,
·     plasma membrane repair, 
·    cell signaling
·     waste disposal system –lysososme is capable for digestion of unwanted materials in the cytoplasm, both from outside the cell and obsolete components inside the cell.