@Lemon's answer is exactly right. Just adding a few points.
The earth's orbit around the sun is due to gravity. Newton explained gravity as an inverse square force law, and F=ma, and almost all you can measure in the solar system due to gravity can be explained with his laws. Einstein realized that it was more than that, that Newton's law of gravity was an approximation when gravity is not too strong and bodies are going slow compared to c, the speed of light. And figured out that gravity is due to the curvature of spacetime.
The Einstein equations allow us to find the more exact answer, with the earth in reality moving in a 'straight' line in the curved spacetime. It turns out that orbit, called a geodesic in general relativity (GR), is the straighest possible curve, the one that minimizes a measure of spacetime distance, in the curved spacetime formed by the Sun's gravity. And to a good approximation it is the same as Newton predicted. But for some effects it predicted slightly different numbers, which were all confirmed.
The SAME gravitational law also predicts gravitational waves when the masses are very dense and heavy, and close, and they move asymmetrically with respect to each other. The gravitational waves emitted by the earth going a around the Sun exist, but are too weak to be detected. You need more massive and dense stars, like neutron stars and blacks holes rotating around each other. That is what we saw in the gravitational waves detected from the two merging black holes - they first rotated around each other, came closer, starting merging and finally merged into 1. In the gravitational wave we detected we saw all of that. Gravitational waves are covered nicely in Wikipedia at https://en.wikipedia.org/wiki/Gravitational_wave
So what is a gravitational wave: it is simply propagating changes (@Lemon called them ripples, same thing) in the curvature of space time. The spacetime shows waves, ripples, propagating through the curved spacetime, making it oscillate as it propagates. It is the same gravitational field predicted by Einsteins equations, it just predicts that when the sources have some asymmetric motions close to each other the mutual effect is to create changes in the overall field, which travels through space over time.
So they are simply different gravitational effects, one a semi-static or slow orbit of one object around another is a quasi-static gravitational field that causes relatively stable orbits, the other a quickly changing field that then propagates. Both come from the same equations, and are represented as different kinds of changes or shapes of the spacetime curvature.
Yes, it travels at exactly the speed of light. And so does any change in a gravitational field. If a new planet suddenly appeared or the Sun suddenly disappeared we would not see them, nor would there be a change in the gravitation we felt, until the time it took the effect to travel here at the speed of light. (You could try to test the speed of grav waves being c, but you'd have a hard time testing the Sun disappearing). They used a rough triangulation of the gravitational wave assuming the speed of light, to roughly locate the patch of sky they came from (it was very approximate, they'll be able to get better accuracies with longer baselines in the interferometers and more of them).
On gravitons, probably a quantum theory of gravity will have to include gravitons, but till we figure out that theory we don't know for sure. All points to there probably being gravitons as the carriers of the gravitational waves, but we don't really know yet what that means. The linearized theory of gravitational waves on a given background spacetime looks the same as a theory of massless particles of spin 2. We've just not been able to successfully quantize the full theory, or detect any single particle that acts like a graviton would - they'd be pretty weak and hard to detect.
By the way, a neutron star is not much more massive than the Sun, it is just about the same mass in a lot less volume. If it was at the edge of the solar system it would have little or no immediate effect on us, though it would affect the outer planets, and over time that could affect our orbit and the stability of the solar system. It's complex dynamics you'd have to figure out.