Conservation of Energy

It's hard to think of a fundamental scientific concept with more explanatory power than energy. It plays a key role in chemistry, biology, geology and even economics as well as physics and astronomy, where it originated. The word, or its Greek precursor appears in the work of Aristotle, but the modern concept of kinetic energy derives from the vis viva of Leibnitz (mass x velocity squared), which Leibnitz believed to be conserved. Leibnitz also anticipated the modern notion that energy lost to friction showed up as heat. Another century and a half or so was needed before the relationship could be proven and clarified.

The power of the idea stems from the fact that energy is conserved. Considerable refinement and redefinition was needed to conserve the notion of energy conservation, though. New forms of energy had to be named and noted: gravitational potential energy, thermal energy, chemical energy and electromagnetic energy. The accounting proved well worthwhile, of course, since the different forms of energy could be converted into each other, and energy represented the capability to do work.

Following the history of that energy allows us to unravel much of the workings of the world. Here on Earth, most energy comes from the Sun in the form of photons of electromagnetic energy, some of which is converted by photosynthesis into chemical energy stored in carbohydrates , which in turn gets converted into that ubiquitous gasoline of life, ATP which powers muscle, brain, and other metabolism.

It's the conservation of energy that proves the most important constraint on life and civilization. The World would be much different if we could build a perpetual motion machine of the first kind and get energy for free. That's why it's kind of shocking to the physicist to realize that there really does seem to be such a perpetuum mobile - the Universe, driven by by dark energy. Sean Carroll has a brief account, but if you want more of the details you really want John Baez here. While Sean is pretty categorical, at least in his post title ("Energy is Not Conserved"), John has a bit more measured take:

Is Energy Conserved in General Relativity?
In special cases, yes. In general -- it depends on what you mean by "energy", and what you mean by "conserved".

Each explains why those two statements don't really contradict each other, but John has more details. Sean, though, explains more of the the role of the dark energy.