Gravitational waves are such things that happens to space when something that has mass changes the direction or speed in which it is moving.
One way to conceptualize them is to imagine a block of wood floating in a pond. If you grab it and move it, ripples will spread out in all directions. These ripples distort the surface of the pond as they pass over it, and when they have passed the surface goes back to being flat again. This is not an exact analogy, but it will give you an idea of how the phenomenon works. In real life, gravitational waves pass through 4-dimensional spacetime rather than across a 2-dimensional pond. And instead of moving space back and forth, they compress and expand it. So when a gravitational wave passes an object, the space it occupies expands and compresses a teeny tiny bit, oscillating back and forth for a short while. The size of the expansion and compression is much less than the radius of an atomic nucleus, so it is very hard to detect.
Gravitational waves are sort of like electromagnetic waves, which are what happens to space when something that has an electric charge changes the direction or speed in which it is moving. They both move at the maximum speed anything can move in our universe, the "speed of light". The big difference is that gravitational waves are more than a trillion trillion trillion times weaker than electromagnetic waves. That's 36 orders of magnitude. Nobody knows why the gravitational force is so much weaker than the electromagnetic force, but that is a fundamental fact about the universe. A single magnet small enough to hold in your hand can generate enough electromagnetic force to lift up a paperclip against the entire gravitational force generated by the planet earth! That's how much the two forces differ.
Although it is trivially easy to detect electromagnetic waves (your eyes do it all the time, because light waves are one kind of electromagnetic wave) it is very very difficult to detect gravitational waves because they are so much weaker.
Why are they important? Everything we know about the universe so far comes from studying electromagnetic waves. We use telescopes that can receive and record different kinds of electromagnetic radiation: visible light, infrared, ultraviolet, radio, etc. This has told us an enormous amount about the rest of the universe, but it has one big limitation: there are lots of things out there that get in the way. Stars, galaxies, clouds of gas, etc. all emit and absorb electromagnetic waves, so we cannot easily "see" what is on the other side of them. It is difficult for us to see things very far away, because the further away something is, the more likely that its electromagnetic radiation has been absorbed or interfered with by things that lie between it and us.
Gravitational waves, on the other hand, are not absorbed by anything. So if an event occurs somewhere in the cosmos that emits a powerful gravitational "ripple" (such as two black holes colliding, which is probably what caused the waves whose detection was recently announced) we will be able to "see" it no matter what else is in the way. And this is important, because in astronomy distance = time. The further away something is, the more time has passed since the waves we are seeing now were emitted. So the farther away we can look, the farther back in time we can look.
Now that we have demonstrated that we can detect gravitational waves, efforts are under way to build even more powerful instruments that can detect weaker "ripples", and this will give us a powerful new means of figuring out what has been happening in the far reaches of the cosmos and far back in time.
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