In 4 dimensions, it takes 20 numbers to specify the curvature at each point. 10 of these numbers are captured by the "Ricci tensor", while the remaining 10 are captured by the "Weyl tensor".

Recall the definition of the Ricci tensor in terms of coffee grounds floating through outer space.

We consider a bunch of initially comoving coffee grounds near a point P in spacetime, with the coffee ground that actually goes through P having velocity v at that instant. (Hence the term "instant coffee".) Working in the local rest frame of the coffee ground that goes through P, we consider a small round ball of comoving coffee grounds centered at P, and see what happens as time passes. Each coffee ground moves along a geodesic, but since spacetime is curved, the ball may shrink, expand, rotate, and/or be deformed into an ellipsoid.

The Ricci tensor R_{ab} only keeps track of the change
of volume of this ball. Namely, the second time derivative
of the volume of the ball is -R_{ab}v^{a} v^{b} times the ball's original
volume. The Weyl tensor tells the REST of the story about
what happens to the ball. More precisely, it describes how the ball
changes shape, into an ellipsoid.

Ted Bunn interjects:

"I've never been too clear on exactly what information is contained in the Weyl tensor.I can say one thing that may help, though. It is true that the Einstein equation only contains ten pieces of information, although you need 20 to specify the curvature tensor. So the Einstein equation doesn't let you reconstruct the complete curvature tensor. That sounds disturbing at first, but it's really OK.

As with many things in general relativity, it can help to state the corresponding fact about electricity and magnetism. If you know the charge and current distributions everywhere in space, you might think that that would let you figure out the electric and magnetic fields everywhere. But it doesn't. There's extra information in the fields beyond just what the sources of the fields can tell you. After all, you could have an electromagnetic wave passing by. It needn't have any source, but it still alters the fields.

So in electromagnetism, knowing all about the sources isn't enough to specify the fields. In general relativity, knowing all about the sources (the stress-energy tensor T) isn't enough to tell you all about the curvature. In both cases, you can supplement the source information with some extra initial conditions to get a unique solution. (For example, in electromagnetism you can specify that no electromagnetic waves are zooming in from infinity. That's enough to give you a unique solution to the fields given the sources. For general relativity, you can perform similar feats, although it's technically trickier.)

Anyway, I hope that makes it a little bit clearer why people say that the Weyl part of the curvature has to do with gravitational radiation: the Weyl tensor carries information about the kind of curvature that's independent of the source distribution, sort of like electromagnetic waves are fields that propagate independently of whatever sources are around."

That makes sense! When we are in truly empty space, there's no Ricci curvature, so actually our ball of coffee grounds doesn't change volume. But there can be Weyl curvature due to gravitational waves, tidal forces, and the like. Gravitational waves and tidal forces tend to stretch things out in one direction while squashing them in the other. So these would correspond to our ball changing into an ellipsoid! Just as we hoped.

Similarly, when a ball of coffee grounds falls freely through outer space in the earth's gravitational field, it feels no Ricci curvature, only Weyl curvature. So the "tidal forces" due to some coffee grounds being near to the earth than others may stretch the ball into an ellipsoid, but not change its volume.