The organelle primarily responsible for the breakdown of lipids within a cell is the lysosome.

Here's why:

* Lysosomes are membrane-bound organelles that contain a variety of hydrolytic enzymes. These enzymes can break down a wide range of molecules, including lipids, proteins, carbohydrates, and nucleic acids.

* Lipase, the enzyme specifically responsible for breaking down lipids, is found within lysosomes.

* The process of breaking down lipids in lysosomes is called lipolysis.

While other organelles play a role in lipid metabolism, the lysosome is the primary site of lipid breakdown within the cell.