Yes, there is a roughly finite amount of oil in the world. (Of course more oil is produced each year as long dead organic matter is slowly converted into more fossil fuel deposits, but that process is MUCH slower than our use of oil.) But the problem that many environmentalists and peak oil worriers have is that they overstep and make incorrect claims based on that basic insight. See Cheney's statement (although he's obviously not an environmentalists). According to the above, Cheney claimed peak oil will hit in 2003. That was obviously false since global oil production in every year since 2003, except 2009, has been greater than that 2003.
See also the classic bet of John Holdren & Paul Ehrlich vs Julian Simon. Holdren & Ehrlich are somewhat prestigious scientists, and Simon was a rather pedestrian economist. But Holdren & Ehrlich lost the bet on the future price of various scarce metals, and Simon won. Why? Because prices, supply, demand, etc., are determined by political economy in the short and mid-term. Regarding oil, strange as it may sound, the earth will almost certainly never run out of oil. Why? Because as it gets scarcer & more expensive, people will switch to another fuels.
SO WHAT TO DO?
Let's not concern ourselves with scarcity, with running out. Let's look at environmental impacts. Let's look at wars perpetrated to control oil. Let's look at the human toll. Let's look at jobs. We can move to renewables and energy efficiency without relying on the peak oil argument.