Galileo was, perhaps, the first to experiment in any serious way with gravitational theories in the late 16th/early 17th centuries.  While the tale of him dropping balls from the top of the Tower of Pisa is almost certainly apocryphal, he did perform experiments using balls rolling down an incline to show that they all rolled at the same rate regardless of composition and weight.  Perceived wisdom at the time, based on Aristotle, who managed to get just about everything to do with physics completely wrong, was that the heavier ball would accelerate faster.  Gravitation theory really got going with Newton's "Law of Universal Gravitation".  This was radical at the time, and has proved to  be remarkably accurate in most circumstances.  After all, when NASA sends a probe into space, they use Newton's Laws to plot its  course.

General Relativity changed the way we look at gravity by describing it as a consequence of the curvature of spacetime.   In real terms, this gives the same results as Newton except at the extremes of relativistic velocity, or in very large  gravitational fields like those around a neutron star or black hole.  Along with the electromagnetic force, gravity is the force  with which we all interact constantly.  It keeps our feet on the ground, maintains the Earth orbiting the Sun, and it is  responsible for the tides. 

As a quantum field theory, the Standard Model has been successful in describing the other three forces of nature with particles called Bosons, so the graviton is the hypothetical bosonic particle that mediates the force of gravity.  Thus, as the  electromagnetic force is the result of exchanging virtual photons, gravity is the result of exchanging virtual gravitons.  This  contrasts to the way that General Relativity describes gravity as the curvature of spacetime.  The graviton has no mass, and has  spin 2, unlike the other force mediating particles all of which have spin 1.  Remember; all bosons have integer spin.  When  gravity is described in this way, it reproduces General Relativity at normal distance and energy scales, but fails at the small distances and high energies described at the Planck scale.  This would be resolved only by a cohesive formulation of quantum  gravity.  General Relativity is a back-ground independent theory, while the Standard Model is not.  This makes any  reconciliation even more difficult.  Certainly, whether a theory of quantum gravity should be back-ground independent is  unclear.  One reason for the popularity of string theory is that it dispenses with these infinities, and it also includes a natural massless, spin 2 particle.  On the other hand, string and "M" theory are not fully back-ground independent, so many scientists lean towards theories like Causal Dynamical Triangulation and Loop Quantum Gravity as they are background independent. 

Gravity is by far the weakest of the four forces.  The other three forces have strengths that vary on a scale of about one million, with the strong nuclear force being the strongest.  Gravity is a factor of 6x10-39 times weaker!  This is known as the "hierarchy problem".  Consider, for example; a small magnet can lift a paper clip (via the electromagnetic force) against the pull of the entire Earth (gravity)!  This is one of the mysteries of gravity.  Unfortunately, having such low energy, it is almost impossible to detect gravitons.  Identifying a graviton is not possible at anything near the energies we have available.  One way to confirm the theory is by detecting gravity waves, which are essentially the manifestation of a vast number of gravitons.  Detecting these would give us insight into the nature of the particle.  There is indirect evidence that gravity waves exist.  Binary pulsars, for example, have been observed to loose energy and spiral in towards each other, in a way that is predicted by General Relativity as a  consequence of radiating gravitational energy; please refer to the section on General Relativity

There have been three confirmed gravity wave detection events detailed by the LIGO project.  Click here for details