Probably the single most important thought in human history:

A little over 2000 years ago, Euclid gathered up the thoughts of his fellow Greeks on the topic of geometry into a coherent book, which attempted to prove everything about the subject from five little axioms. Euclid's idea was that by choosing a few axioms which were so obvious no one could argue against them, and using them to prove statements about his world, he could be absolutely certain about those statements, and no one could possibly argue against them either. The problem was that one of his axioms wasn't so obvious, or at least some people didn't think it was. There were other issues, regarding minor things which Euclid tacitly assumed, and which therefore had to be stated (and were stated, but only a couple thousand years later) for the proofs to be complete. But those aren't that important right now. What are important are his axioms, which are as follows:

1. Any two points can be joined by a straight line.
2. Any straight line segment can be extended indefinitely in a straight line.
3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
4. All right angles are congruent.
5. If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.

Can you guess which one is problematic? (Hint: It' s the fifth one.) Now, most people would probably agree with Euclid's fifth postulate, especially if they've taken high school geometry. It seems like it should be true, and they're teachers told them that it was true, and they're teachers don't lie to them, so it must be true. Unfortunately, your teachers do lie to you. Not much, and not in a big way, but they will most certainly tweak the truth a bit to make it easier for you to understand something, assuming that if you really need to know the (often significantly more complicated) truth, you'll learn it later on.
The problem is that if you're looking for something so simple and so obvious that no one could possibly argue with it, something that seems right and that most people agree with, simply isn't good enough. Unfortunately, the fifth postulate was integral to Euclidean geometry, and plenty of people weren't willing to throw the project out completely. So someone had the idea, "What if we could prove the fifth postulate from the others?"
People tried. It was difficult, but people did come up with proofs. But every time someone came up with a proof, someone else realized that they were assuming some other postulate implicitly. Moreover, each of these other postulates was logically equivalent to Euclid's fifth postulate, meaning that, given the first four axioms, you could prove Euclid's fifth postulate if you assumed their implicit postulate, and you could prove their implicit postulate if you assumed Euclid's fifth postulate. Basically, the two are interchangeable, one is just a reformulation of the other. This is certainly interesting, but gets us nowhere in terms of proving Euclid's fifth postulate from the other four.
You can find a list of statements which are equivalent to Euclid's fifth postulate on Wikipedia. The most famous of these is known as Playfair's axiom: "Through a point not on a given straight line, one and only one line can be drawn that never meets the given line." Because of this formulation, which is significantly simpler than Euclid's original axiom, Euclid's fifth postulate is now often known as the parallel postulate. In fact, most modern formulations of Euclidean geometry use this formulation for reasons which should become clear momentarily.
This went on for the next two thousand years. Think about that figure for a second. This was an unsolved problem in mathematics for two thousand years. And it didn't get pigeonholed for a few centuries either. Even when Europe was deep in it's dark ages, Arab mathematicians were still trying to prove it. And still no progress was made until an Italian monk named Giovanni Saccheri decided to pursue an idea (no this isn't the all-important one, we're getting there) originated by Persian polymath Omar Khayyam. His idea was to try to prove Euclid's fifth postulate by contradiction. That is, he wanted to show that if you assumed that the first four axioms were true, and if you assumed that the parallel postulate was false, you would necessarily reach a contradiction. This would in turn show that if the first four axioms are true, the parallel postulate must be true as well, and he'd have a proof.
There are two ways the parallel postulate might be contradicted. The first is to assume that there are no non-intersecting lines. However, from this it can be proven that straight line segments cannot be extended indefinitely, in contradiction with the second postulate. (This is the foundation of elliptic geometry. See below.) The second is to assume that there is more than one line passing through a point not on the given line, which does not intersect the given line. Omar had not been able to find a contradiction given this, but Saccheri thought he might be able to.
Saccheri spent many years exploring the implications of contradicting the parallel postulate, and found that all sorts of wild theorems could be proven, such as the fact that rectangles could not exist, all of which seemed completely absurd to Saccheri, but still he couldn't prove the parallel postulate and he eventually gave up.
That was in the early 18th century. Almost a century later, his work was picked up by a Mathematician named Beltrami. Beltrami saw his work for what it really was - a completely different geometry. Hyperbolic geometry, as it is now known, is a geometry separate from Euclidean geometry, and incompatible with it, in the sense that it assumes the contradiction of one of the axioms of Euclidean geometry, so they cannot both apply to the same world.
But this was the big idea, possibly the most important idea ever: that mathematics was not about studying our world, but about studying possible worlds. Euclid's fallacy, and the fallacy of all of those who tried to prove the fifth postulate, was to assume that these were statements about the world we live in. If you assume that, then you can never prove anything, because you are always working off an assumption which may be wrong. The first four may seem accurate, but there is no way to know absolutely that they are, and so there is no way to be sure that anything you prove from them is true.
But, if we take each of these systems as interesting systems in and of themselves, regardless of whether or not they apply to the world we live in, then we can be certain that anything we prove from these axioms is true within the respective axiomatic system. This idea freed the thoughts of the mathematicians of the 19th century to come up with even more abstract ideas, such as Riemannian geometry, which is neither Euclidean nor hyperbolic nor elliptic (which assumes the contradiction of both the second and fifth axioms). In fact, thanks to Einstein and his professor Minkowski, we know (or are at least fairly certain) that the parallel postulate does not apply in our world, but that the presence of matter or of energy curves spacetime in such a way as to make straight lines (or geodesics, as they are called in Riemannian geometry) curve inward, over time, towards the center of whatever is creating the gravitational field.
Why is this idea so important? Without it, we wouldn't have differential geometry, or general relativity. There might not be any quantum mechanics, or at least not in the way we know it. And of course from quantum mechanics comes modern chemistry, and from that modern biology. How could we ever have discovered the structure of DNA without understanding X-ray diffraction? How could we have gotten as far as we have in quantum mechanics without Hilbert spaces?
This idea, which has revolutionized mathematics, and even, more recently, physics, and has led to so many developments in so many other fields, this, I would say, is probably the most important idea in human history.