Peak Oil is the theory that states that the production of oil (and really all fossil fuels) will achieve a maximum amount at a particular point in time. This maximum will be followed by a prolonged decline characterized by increased energy prices, energy shortages and generalized economic stagnation derived from the first two.
There have been multiple theorists who have given dates for the fulfillment of the theories' prediction. They range from 1979 to 2060. The truth of the matter is that it is very difficult to predict due to new oil findings, unconventional oil developments such as shale oil and tar sands, the development of alternative energy technology, and the increased energy efficiency of modern appliances. However, in spite of all of this, there is the other truth of the matter: it will eventually happen.
When Peak Oil occurs, we would probably not notice. It will be a rear view mirror event. Oil will not go away, it will become more expensive. It is my opinion that there has to be directed efforts to harvest energy aside from oil or from fossil fuels. So the real question for the future should be: When did Peak Oil occur?