Einstein's general theory of relativity states a lot of things about our universe, many of which completely shifted our understanding of it. One of the consequences of this theory is that space and time aren't separate entities; they are instead products of the same thing, called space-time. Space-time is able to be affected by mass, which means that a massive enough body capable of bending space and time itself. It follows that if two massive bodies orbit around each other, they would be able to produce "ripples" in space-time (illustrated below). These ripples are what's known as gravity waves, and with them, we can learn an abundance of things about our universe previously unknown to us.
An artist's interpretation of gravity waves.
Gravity waves share many similarities with waves we're typically used to, save for a few exceptions. Instead of propagating through a physical medium, they instead propagate through space-time in all directions from the source. This allows the waves to travel unobstructed by anything else in the universe, excluding for other gravity waves. Now although measuring gravity waves sounds promising, it has a few caveats. For one, gravity waves get weaker as they spread farther from the source, much like typical waves do. They are also very far from us, which usually means that by the time a wave reaches us, it is much, much smaller than how it started. The size of a gravity wave when it reaches us tends to be in the range of 10^-18 meters, or about 1000 times smaller than the diameter of a proton (LIGO.org)! Gravity waves are also very hard to detect. How do you measure something that affects all of space and time (including all telescopes/detectors) on a very tiny scale? 
To answer the previous question, the way scientists go about this is through interferometry. In its most basic form, an interferometer consists of two mirrors, a beam splitter, a laser, and some very sensitive detectors. With these, they can shoot a beam of light, split it into two (or more) directions, and measure the time taken from each beam to reach the detector. Since gravitational waves distort space, when a gravity wave "hits" the interferometer, the distance between the mirrors and the detector gets shifted by a very slight, but detectable, amount. This shift allows us to calculate how large the wave was, and from there work out what caused it. Since gravity waves are very minute, we are only able to detect waves produced by that of very massive, very violent events. For example, many detectable gravity waves come from collisions of two black holes or neutron stars, supernovae, or from two very massive objects (neutron stars, black holes, supermassive stars) orbiting each other very fast. Yet, even in these cases, the gravitational waves are still small, and need very precise instruments to measure. Enter the Laser Interferometer Gravitational-Wave Observatory (LIGO). LIGO is the first gravitational wave detector that has been able to accurately measure gravitational waves, which has opened the doors for an entirely new field of space observation. Although it is relatively new, we have gained much insight into our universe already, and the extra implications make gravity wave detection a promising field.
An example of a laser interferometer
The detection of gravity waves has allowed us to witness things that we wouldn't be able to see otherwise. For instance, as previously discussed, it allows us to detect things like black holes, and the collision of neutron stars. From it, we've already learned a lot, such as that neutron star collisions are mostly responsible for many heavy elements in our universe, such as gold and platinum. It has also helped in proving Einstein's theory of general relativity. Scientists are hoping that it may allow us to explain things like the nature of gravity, or even events that happened near the Big Bang. It's tough to say just how much we can learn from gravity waves, but with time we may gain a deeper, more fundamental understanding of our universe through them.
References
forbes.com/sites/startswithabang/2017/10/16/astronomys-rosetta-stone-merging-neutron-stars-seen-with-both-gravitational-waves-and-light/?sh=ca8c85e17255
imagine.gsfc.nasa.gov/science/toolbox/gwaves1.html?linkId=99587704
ligo.caltech.edu/page/what-are-gw
ligo.caltech.edu/page/what-is-interferometer
ligo.org/science/GW-GW2.php
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