Type I a Supernova.
Scientists have known that the Universe is expanding for a long time, as it was first observed in 1929. This dispelled the previous pervading knowledge that the Universe was static, staying the same size forever, with no beginning or ultimate end. The discovery of an expansion also implied that at some point in the past, the Universe was unimaginably dense, until the "Big Bang" caused it to first expand, and the expansion of that event has been carrying it outward ever since then. However, the prevailing knowledge for decades after this discovery was that, although the Universe was expanding even since the Big Bang blew it apart at the very beginning, it was gradually slowing down due to the gravity of everything inside it. This all changed in the 1990's, specifically in the year 1998, when two independent research teams in the field of astronomy uncovered clues that led them to believe that the expansion of the Universe was in fact accelerating. Later, in 2011, the Nobel Prize in physics was awarded for proving that the Universe's expansion was, in fact, speeding up. This has large consequences for the future, as now the Universe has no chance of its gravity overcoming the expansion rate and collapsing back down again. This means that it would eventually get so large, and the rate of expansion so fast, that galaxies would be permanently isolated from each other in the distant future.
This discovery was made by looking throughout the distant Universe for supernovae. This was driven by the desire to find out how far away these distant objects in the Universe were, which is useful because it can tell how old they are as well. The reason supernovae are so important for this is because supernovae, specifically a type called Type I a, are always produced by a white dwarf star gaining too much mass and collapsing in on itself before blowing up -- an event that emits the same brightness of light, no matter where it is in the Universe. This means that by measuring its apparent brightness from Earth, we can use them as standard candles to determine precisely how far away these supernovae are from us. Also, we can compare the numbers we get from measuring supernovae with how much these galaxies are redshifted from us -- how quickly the expansion of the Universe is driving them away from us -- and find out an even more accurate and insightful figure for the measurement by comparing these two techniques, red shifting and supernova spotting.
But these teams of astronomers found something very strange when they did this. They found that the supernovae were too dim to be the distance from us that red shifting alone would suggest, meaning that the expansion of the Universe can't be slowing down or remaining constant -- it must be getting quicker instead. This was ground shaking news for the scientific community. It had taken decades for this result to be discovered, mostly because the idea of all the matter in the Universe gravitationally attracting to each other and overcoming the expansion at last made so much sense, and because to properly measure these supernovae, and therefore conclude that the Universe must be expanding faster every second, requires significant advancements in technology that weren't available until the 1990's. Still, this was definitely worthy of a Nobel Prize, for overturning an age-old paradigm about the nature of the Universe itself.
I personally feel that this is enough evidence to conclusively say that the expansion of the Universe is accelerating, but many people, even scientists, still find it hard to believe that with all the gravity pull the Universe together, it will eventually be doomed to fly apart at incredible and ever-increasing speeds. But the reason I can accept it is because the same thing occurred when the steady-state theory was disproven in the 1920's, and the fact that the Universe was expanding at all was discovered. People were skeptical of that and didn't believe it at first, but eventually it was accepted into the mainstream as scientific fact. And hopefully this new discovery of the acceleration of the expansion of the Universe will be completely accepted too, even though it is honestly very hard to wrap our mortal heads around, especially the far future implications of it.
One more thing about this discovery that is worth mentioning is that there needs to be a very good explanation for why the collective gravity of all the galaxies in the Universe doesn't just slow the expansion instead of speeding it up. There has to be some invisible force, driving galaxies away from each other at ever-increasing speeds. In fact, scientists believe that it isn't the galaxies that are being accelerated by this force, but that space itself is being inflated by it, which is by far the weirdest thing so far. One reason that this is the case, not that the galaxies are physically moving away, is that the acceleration is completely unbounded, and it will eventually far surpass the speed of light, which is only possible if it is space itself that is moving at faster-than-light speeds throughout the entire Universe. But what is actually driving this incredibly rapid expansion? Scientists don't know, but because it solves a great problem in cosmology, similarly to Dark Matter with galaxy spin rates, they have dubbed this unknown phenomenon "Dark Energy", and they still have no clue what it is made of, or where it comes from, or where it goes. They do know one thing, however -- whatever this thing is, it takes up around three quarters of the Universe! This means that, what with Dark Energy and Dark Matter, all the "normal" matter in the Universe isn't so common, or normal, in the grand scheme of things, after all....