The graviton is a hypothetical elementary particle that transmits the force of gravity in most quantum gravity systems. In order to do this gravitons have to be always-attractive (gravity never pushes), work over any distance (gravity is universal) and come in unlimited numbers (to provide high strengths near stars). In a quantum theory this defines an even-spin (spin 2 in this case) boson with a rest mass of zero.
Gravitons are postulated simply because quantum has been so successful in other fields. For instance, electrodynamics can be very well explained by the application of quantization to photons. In this case photons are being continually created and destroyed by all charged particles[?], and the interactions between these photons produce the macroscopic forces we are familiar with, like magnetism.
Given the widespread success of quantum in describing the vast majority of basic forces in the universe, it seemed only natural that the same methods would work well on gravity as well. Many attempts were made to introduce a so-far unseen gravition, which would work in a fashion somewhat similar to the photon. It was hoped that this would quickly lead to a quantum gravity theory, although the maths was a bit difficult.
It has not worked out that way. Any such theory would require a graviton to operate in a fashion similar to a photon, but unlike electrodynamics where the photons act directly on each other and their charged particles, gravity just doesn't work so simply. Well observed behaviours show that gravity is created by any form of energy (mass simply being a particularily condensed form), which is difficult to describe in a fashion similar to "charge". To date all attempts to create a consistant simple quantum gravity theory have failed.
Detecting a graviton, if it exists, would prove rather problematic. The particles carry very little energy1 so detecting them would be very difficult. The only way to detect them would be to look for cases where the overall motion or energy of an object changes in a way that is different than predicted by general relativity, but one of the basic principles of quantum gravity would be that it matches those predictions as closely as possible.
It should be noted that a quantum gravity theory does not require a graviton, loop quantum gravity has no analagous particle for instance.
1) Many people are surprised to learn that gravity is the weakest force. A simple experiment will demonstrate this however, lift your arm. You have now lifted several kilograms of mass against the gravity generated by the entire planet.