(Note: this is a rather terse and disjointed first take on my outlook. I hope to expand it in the future.)
The phrase 'Peak Oil' is term that means many things to many people, like its close cousin Global Warming, though far fewer people have heard of Peak Oil. It can be many things because it is not a discrete entity like clouds, or humpback whales, or lunch. In the minds of the various people who know the term it is a varying composite of engineering, science, and culture - whether it should be or not.
Peak Oil is first and foremost a physical phenomenon that can be observed, and has been repeatedly by oil industry engineers. The rate of oil production in field after field has followed what to the casual observer would appear to be 'bell curve' when plotted out visually. The flow of oil starts off slowly, then enters a period where it rises quickly to a rounded peak, followed by rapid decline, and a final period of minimal extraction. The curves of individual fields vary slightly for a variety of reasons: political turmoil, accidents, poor management, and more. Technological developments also tends to make the decline side of the curve less steep due to better recovery. However, all fields eventually decline, usually sooner than later.
Science enters the Peak Oil concept through explanations of the phenomenon. Oil and other fossil fuels are formed by geological process that scientists have explained fairly well. Science also explains why fossil fuels are relatively rare and are not forming at rates anything close to the rates at which they are being depleted. Science also explains why various fields produce at different rates, have different mixtures of oil and gas, and so on.
Finally culture rears its head when the implications of Peak Oil are contemplated. To some it seems like the end of the world, and say so. Others, overwhelmed by the implications for the lives that they currently live, discount it entirely, preferring to listen to those who attack it for baser, more selfish motives. To yet others it is some kind of nefarious plot dreamed up by unknown people aimed at destroying this or that. In other words, Peak Oil generally either slots easily into a persons existing mental framework, or gets rejected because it does not. For a few (but hopefully growing number of people) coming to understand Peak Oil as a observable phenomenon changes their mental outlook, as well it should.
I am definitely a person whose outlook has been changed by coming to understand past patterns of oil discovery and production, likely future discovery and production, and the implications of a serious decline in oil production, along with production of other fossil fuels. These amazingly concentrated sources of energy have enabled modern human civilization. Without these precious strokes of geological luck development would have slowed significantly in the late 18th or early 19th century as biomass-based energy resources became depleted. With fossil fuels humans have been able to radically reshape their lives and the whole earth, for better or worse. And now our frenzied development has reached a point where it could harm ourselves, unless changes are made.
The sciences, in addition to explaining the how sources of energy arise, give us an important rule: energy can be used only once. (This is actually an implication of the second law of thermodynamics, not a formal law in itself.) If energy is readily available, it can be used with abandon. But if concentrated sources of energy are rare, then great care should be taken when using energy to make sure as much as possible is going into useful work. Peak Oil implies that concentrated energy sources are rare and likely to decline at current usage rates. Unfortunately, modern patterns of living developed when the amount of energy available appeared or were reported to be nearly limitless.
Unless a major new source of energy is discovered, a significant new energy technology is developed, or significant mitigation efforts are undertaken, the implications of Peak Oil for human civilization are profound and not all that appealing. At this point in time the chance of finding major new fossil energy deposits is low, and even if they were discovered utilizing them would just add to the other major threat man-made facing the world - Global Warming. The chance a major new energy technology coming on-line in the near future, such as nuclear fusion, is also low. That leaves mitigation, both in the form of reducing energy use, and making major coordinated investments in existing energy technologies such as solar, wind, and nuclear fusion. Investments should be both in new plant as well as incrementally improving these existing technologies.