This is a commonly asked question and has many people scratching their heads. The difference between these names is a matter of how strongly concentrated the oils are in the fragrance.
Parfum contains the the highest concentration of perfume oils, followed by Eau de Parfum, then Eau de Toilette. The higher the concentration of oils is in your fragrance, the longer the smell will last on your skin and the smaller the amount that is needed to be applied.
Although it depends on your skin's chemistry, Parfum is an intense concentration and should well and truly last all day. Eau de Parfum usually lasts up to 8 hours and Eau de Toilette has around 4 hours of wearability.
You will probably find that your favourite fragrance comes in these different concentrations. Many people like to wear stronger concentrations, such as Eau de Parfum in the cooler months but find a lighter version or Eau de Toilette more enjoyable and refreshing to wear in the sticky summer months.