Final answer:
Peroxisomes are the organelles that break down fatty acids and contain the enzyme catalase, which degrades hydrogen peroxide into water and oxygen.
Step-by-step explanation:
The breakdown of fats and the presence of the enzyme catalase are primarily associated with peroxisomes. These small, spherical organelles enclosed by single membranes carry out oxidation reactions that break down fatty acids and amino acids. In addition to these functions, they are responsible for detoxifying poisons that may enter the body, a common example being the detoxification of alcohol in liver cells. Peroxisomes contain catalase, an enzyme that degrades hydrogen peroxide (H₂O₂) into water and oxygen, to prevent cellular damage from this byproduct of oxidation reactions.