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. They should also not be confused with the unrelated gravity waves.
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 carried by a particle, then it is also due carried by a wave!
This is one of the great remaining mysteries of the universe. I have an idea 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 thought of 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 idea (simplified):
This recent YouTube video by ScienceClic English actually illustrates my idea very well:
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. Since the linked video is explaining Relativity, perhaps my idea is actually an explanation of Relativity.
Several things this explains:
This explains why objects with more mass have more gravity (e.g. each bit of mass is absorbing a small amount of space, so has gravity). The more mass you have, the more gravity you have.
This explains why gravity is strong when near an object (e.g. when close to earth), and decreases inversely proportional to square distance from the object. If you imagine an expanding "spherical wave" of space moving inward, surface area of the sphere increases in relation to the square of the distance. However, the effect is dispersed over this increased surface area, hence it decreases with the square of the distance from the object. A fairly simple (but 2D vs. 3D) analogy is how when you throw a stone into water, the ripple will be strong at first, near the entry point, and then the ripple disperses as it expands into the surrounding pond. A good 3D analogy is how sound (or light) expands spherically, and is more faint the further you are away from the source.
The orbit of "binary" systems, including the dynamics between the earth and moon. The earth is more massive than the moon, and is absorbing more space than the moon. Therefore, it is moving the moon toward it fairly quickly (but the moon has enough momentum that it remains in orbit vs. falling toward earth). However, the moon is also pulling on the earth, so the earth is also "falling toward the moon" a little. Again, the earth has enough momentum not to "fall into the moon". This results in mostly the moon orbiting around the earth, but the earth is also "slightly" orbiting around the moon, which would be observed as a slight wobble.
This explains why large mass objects and small mass objects are affected the same by gravity. Since it is space itself that is moving, whatever is in that space will be moved accordingly. If it is an anvil, it will be moved just the same as a feather. Therefore (if it weren't for other effects such as air resistance), a feather would fall to earth the same speed as an anvil. In practice, air resistance makes the feather small much slower (but repeat the experiment on the moon, and they should hit at the same time since it has no air).
This even explains why things without mass (e.g. light) are also affected by gravity. Since space itself is moving, light moves along with it. This would explain gravitational lensing, and why light cannot escape a black hole.
Second part of my idea (more speculative):
I also believe antimatter somehow constantly ejects space back into the universe. In the video, imagine the reverse happening (e.g. if you could play it in reverse). 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.