"You mention Einstein's description of gravity not as a force between particles but as deformations of their local space-time geometries. Is gravity thought of as "just" a measurement of the space-time deformations? If it is, why would astronomers be looking to detect "gravity waves" or "gravitons" or gravity "particles"?
We'll begin with the first part, which we think is the easier of the two concepts, at least for us. Einstein's General Theory of Relativity describes the "force of gravity" as resulting from deformations that massive objects induce in their local space-time geometries. We remain on Earth's surface because we're trapped within the planet's space-time indentation. The planets are similarly trapped within the Sun's gravity as well as the Sun is trapped within the galaxies. The galaxy is bound to the Local Group, which is within the Virgo Supercluster and so on and so forth. (Gravity is an efficient tyrant, capable of presiding over mountain ranges as well as kitchen sinks.) Whereas Sir Isaac Newton described gravity as a force mediated between all massive objects, Einstein showed that gravity is a condition within space-time, itself.
To understand the concept of gravity waves, we begin with a particularly absurd thought experiment. Imagine that for whatever reason, the Sun suddenly snapped out of existence. It didn't go supernova (it's not massive enough to do that, anyway) nor did it cast off its outer layers in an expanding shell. It just left the Universe in a snap. What would happen to us? Well, for a few minutes anyway, life wouldn't change much. As the Sun is about 8.2 light minutes away, Earth would still receive its light for 8.2 minutes following the Sun's disappearance.
However, what about the gravitational influence? Would the gravity snap off in an instant? No, it would not. The gravitational pull wouldn't stop instantaneously. Instead, the gravity wave communicating the Sun's sudden absence would propagate through space-time at light speed. Earth would remain gravitationally bound for 8.2 minutes until the indentation, as it were, smoothed itself out. At the moment when the gravitational influence stopped, the Sun would appear to vanish and our planet would move tangentially along its orbit. The chances of the Sun actually falling through some hyper-dimensional trap door are somewhat remote, but we use this example to help illustrate the concept of a gravity wave.
Though predicated by General Relativity, gravitational waves have not yet been detected. The problem involves the distances of the bodies capable of transmitting those waves large enough to be detected. Astronomers expect that such waves would be emitted by asymmetric supernovae, those stars whose explosions are not equally powerful in every direction, or orbiting bodies, to name just a couple examples. Ordinary planetary orbits would generate such meager waves that even astronomers in proximate star systems would be hard pressed to detect them. Instead, regard a much more energetic system such as a binary pulsar, consisting of two rapidly spinning neutron stars orbiting about a common barycenter, could produce gravitational waves powerful enough to register even tens of thousands of light years away. According to Einstein's theory, the system should gradually lose energy in the form of gravitational waves. Consequently, their separation distance will decrease and their orbital velocity increase.
The first binary pulsar discovered PSR B1913+16* has exhibited such deterioration. Presently, the mean separation between the components is diminished by 3.5 meters per year, resulting in a 76 microsecond orbital period decrease. These values will gradually increase, causing a predicted inspiral (collision) in approximately 300 million years. Attempts to detect gravitational waves from this system and others have not been successful, even with LIGO, the Laser Interferometer Gravitational-Wave Observatory. However, gravitational physicists are confident that one day these waves will be found, providing further verification of Einstein's already well established General Theory.
Gravitons are another matter.
We should explain first that physicists have identified four fundamental physical forces: gravity, electromagnetism, and the weak and strong nuclear forces. Scientists have determined that each of the last three forces are mediated by elementary particles. When two magnets react to each other, they are exchanging photons. Protons and neutrons are held together in atomic nuclei through the strong nuclear force, which binds them with "gluons." The weak nuclear force, responsible for radioactive decay, is transmitted through W and Z bosons. However, physicists have not yet detected gravitons, elementary particles believed to transmit gravitation. The main problem with this idea is that it seems irreconcilable with General Relativity that describes gravity as intrinsic to the space-time continuum and not as a remote particle exchange. The attempt to reconcile General Relativity with the graviton is the basis of the still elusive "quantum gravity." It is plausible that a gravitational wave could be regarded as a series of gravitons in a similar (coherent) quantum state. As of now, such models are all hypothetical. No gravitons have been detected and it seems highly unlikely that they would be with current detectors, as they would be far weaker than any other elementary particle, if they even exist in the first place.
I hope this answer proves helpful and doesn't make matters worse.
*This is also called the Hulse-Taylor binary pulsar, after Joseph Taylor and Russell Hulse, who discovered it in 1974. They were awarded the 1993 Nobel Prize in Physics for this discovery.