Today's discovery of gravitational waves from the origin of the universe is the most important discovery in science since that of DNA. I'll try explain why. But first, I have to admit that I am not an expert physicist. If you want the technical background go to someone who really knows what is going on, such as Sean Carroll - http://www.preposterousuniverse.com/blog/2014/03/16/gravitational-waves-in-the-cosmic-microwave-background/ I hope Sean writes a book on this, as his ability to explain such complex ideas is exceptional. However, I'll try and give some idea.
Gravity curves space and time, as revealed by Einstein's work on General Relativity. The reason why gravity pulls and diverts objects is because of this curvature. Newton said that unless acted on by a force objects will move in straight lines. Objects in a gravitational field move in the equivalent of straight lines in curvy space - these lines are called 'geodesics'. Think of the surface of the Earth - the most direct route between two points at different longitudes away from the equator is not a straight line, but a curve called a 'great circle'. 'Great circles' are geodesics. Following these 'curvy straight lines' is why objects are diverted by gravity.
As well as describing this curvature of space, Einstein also showed that changes in gravity travel at the speed of light, not instantly. If the Sun vanished, it would take 8 minutes for the Earth to start to leave orbit, for example.
If you have curvature and that curvature can change, and the change has a speed limit, then you can get waves. Suppose the gravity of the Sun could be turned on and off. If this happened then pulses of gravity would travel out from the Sun's position. If you have two objects orbiting each other, the constant change in position of the sources of gravity results in waves.
We know that gravitational waves exist because waves carry away energy and we can predict how quickly this will happen. When the objects orbiting each other are very dense indeed, like neutron stars, then they will lose a lot of gravitational energy as waves, and their orbits will decay, and we have seen this happen.
Gravitational waves in the cosmic microwave background are such an amazing discovery because the cosmic microwave background is a microscope. The microwave background is the light released when the universe first became transparent, when electrons stopped flying around and joined up with atomic nuclei. This happened about 379,000 years after the origin of the universe. The microwave background is extremely smooth, which is strange because the expansion of the universe is so fast that widely separated areas of the background can never have come into contact, to smooth out any differences. So why is it so smooth? One answer is inflation - the idea that the region of the universe that we can see started off extremely small and so was in contact, and then expanded incredibly fast, smoothing out any differences. This expansion was so great that tiny quantum fluctuations became big enough to create the unevenness needed to allow gravity to collect gas together to build galaxies. Subtle fluctuations in the microwave background reveal what went on at a quantum scale. So, gravitational wave effects in the microwave background show us ripples in gravity at a quantum scale, and the only way we can think of that these ripples have become big enough for us to see is if the ripples were blown up enormously because of inflation.
The cosmic microwaves take us back to hundreds of thousands of years after the origin. We can see no further back because before that the universe wasn't transparent to light. But, just about everything is transparent to gravity and gravitational waves. When we see the gravitational waves we are looking back to the instant when our universe began, perhaps around the Planck time: 10-44 seconds - probably the shortest time that makes sense in our universe. This not only confirms the idea of inflation, but it should allow us to get some idea of what happened at the very beginning of the timeline of our universe.