More specifically, the major enzymes in peroxisomes break down purines, which are of of the two major types of nitrogenous bases.
In plants, the peroxisomes are where a specific set of photosynthesis-related reactions take place when cells containing a high quantities of oxygen are exposed to sunlight. These reactions result in the production of hydrogen peroxide, which would be quite corrosive to the cell if it were not for the presence of yet another enzyme in the peroxisomes that immediately breaks the H2O2 down into harmless water and oxygen molecules.
In human beings, peroxisomes play an important role in the body's ability to break down fatty acids, the formation of cholesterol, the production of bile acids, and the production of lipids that help form cell membranes (including myelin, which sheaths the nerves). Peroxisomes are also important in preventing the excess production of oxalate, which can cause kidney stones.
Thus, when a person's peroxisomes fail to function properly, a variety of symptoms can develop. Diseases such as malonic aciduria, pseudo-Zellweger syndrome, Refsum disease, and X-linked adrenoleukodystrophy are all related to peroxisome failure.
Information sources:
Biology by Helena Curtis and http://www.peroxisome.org/