Peroxide Value is an empirical measurement of the amount of oxidation that has occurred to an oil. After the refining, bleaching and deodorization process, an oil has a very low peroxide value, typically less than 1.0 mEq/kg.
Air, or more specifically the oxygen in air, can react with the oil and form various peroxide components that can eventually impart odors and off-flavors to an oil, effecting its quality.
Remember that vegetable fats and oils are compounds called triglycerides. These triglycerides in turn are made up of different fatty acids, some are saturated (having no double bonds) and others are unsaturated (having one or more double bonds). It is these double bonds that react with the Oxygen to form these peroxides.
The rate at which a oil oxidizes and eventually becomes rancid is known as its stability. The stability is directly affected by the level of unsaturation a oil has. Therefore a oil with a high saturated fat content, such as Coconut Oil, are highly stable, while oils with a high degree of unsaturation such as Soybean Oil, are not very stable and tend to go rancid much faster than other oils.
There are other products, known as antioxidants, that will also increase the stability of an oil. These antioxidants prevent the oxygen from reacting with the oil's unsaturation, therefore extending its stability.
So back to Peroxide Value, the lower the value, the fresher the oil. The lower the unsaturation, or higher the saturation, the more stable the oil, and slower that peroxides are likely to form. If someone wants to further extend the stability, antioxidants can be used.