What is gravity? I want to know more than it being simply the "mysterious force" that attracts things to earth. Is it a particle, a wave, or something else entirely?
I can attempt to address the second part of your initial question: "Is it a particle, a wave,...?" Einstein's theory of general relativity states that mass and energy bend space-time. Space-time, in turn, tells matter how to move (John Wheeler put this more elegantly).
This concept is completely different from the theories of the other three fundamental forces (electromagnetism, and the strong and weak nuclear forces). In these quantum theories, forces are mediated by particles called gauge bosons. Electromagnetism is carried by photons, the strong nuclear force by gluons, and the weak nuclear force by W+, W-, and Z bosons. There have been many attempts to find a quantum theory of gravity - that is, to use quantum principles to construct a field theory of gravity. In these theories, gravity would indeed be mediated by a particle, dubbed the graviton. String theory is one example of these theories; the physics community is divided about it.
You probably have heard other terms tossed around that embody interesting concepts. A gravitational wave is essentially a ripple in space-time emitted by an object or system of objects. There are strict restrictions over what sort of objects can emit these waves; binary neutron stars are one consistently cited example. These waves should not be confused with the aforesaid hypothesized gravitons; while gravitational waves carry energy, they do not "mediate" gravity.
Finally, I have heard the term gravity wave used in a completely different context. I can't describe it as well as the other concepts, but I believe it is used to refer to an effect of gravity on other substances. I would advise looking for an answer to that in a textbook. At any rate, it is unrelated to a gravitational wave.
So, basically, the predominant view in the physics community is that general relativity is the best description of gravity; at the moment, gravity is considered to be the bending of space-time, so it is indeed "something else entirely." Many theories of quantum gravity, including string theory, however, attempt to create particles called gravitons as force-carrying bosons. If evidence is found relating to these theories, then we may well learn whether or not gravitons do exist. One more thing about waves: Because of the quantum concept of wave-particle duality, any particle can be described as a wave, which has a wave function. So if gravity is due to a particle, then it is also due to a wave!
This is one of the great remaining mysteries of the universe. I have a theory on this that one can derive known equations for gravity, as described below (this has not been proven, but I believe it is very likely to be the case). Other people have theorized this as well, but it is not yet the mainstream view. In fact, someone wrote a paper on it, discussed here: http://www.angelfire.com/pq/spaceflow/part2.html
My gravity theory (simplified):
I believe gravity is the flow of space itself into mass. By space, I am referring to the 3 dimensions of space that we observe in the universe. I believe all mass constantly "absorbs" the space around it, bringing everything around closer to itself. To conceptualize this, imagine mass being a vacuum cleaner, sucking up the air, and the dust that is in the air all around it. This has the greatest effect on the air that is nearest to the vacuum, and I believe mass behaves likewise with space.
Several things this explains:
Second part of my theory:
I also believe antimatter somehow constantly ejects space back into the universe. I believe when matter and antimatter form, they are somehow "linked" so that space absorbed by matter is ejected by the corresponding antimatter particle. This would give antimatter a "negative" gravity effect. I believe there is equal amounts of matter and antimatter in the universe, and the antimatter is in a diffuse cloud spread throughout the universe, where matter is not present. I believe this accounts for dark energy - the accelerating expansion of the universe.