Galileons arise as a sort of effective field theory description of a very peculiar set of models. So the original idea was you take some 3+1 dimensional Brane that is embedded in some ambient higher dimensional (with the DGP model in mind but not necessarily limited to it) 5d bulk. You then look for effective field theory descriptions by integrating out the bulk space, and you are left with a very specific action that has a scalar that is kinetically coupled with the metric. The scalar encodes much of the residual information about the higher dimensional space, and you find that for consistency (to enforce the Vanshtein effect) you have to have not just the usual shift symmetry acting in field space, but also the gradient shift symmetry . In some sense this symmetry is a relic of the broken 5d lorentz invariance and the broken 5d reparamitrization invariance. However for the purposes of the effective field theory we are to view this as a sort of internal symmetry and forget about where it came from.
Now the interesting thing, is that the ensuing effective scalar-tensor theories inherit much of the unusual properties of the higher dimensional spacetime.
So anyway, the properties of the type of scalar fields that have this symmetry (called Galileons) are unusual, and surprisingly rich, and the ensuing program has been to study the detailed phenomenology and to fully catalog the type of modifications of gravity that are possible. What's fascinating is the Galileon symmetry enforces terms that are no more than 2nd order in derivative in the equations of motion and thus avoid the usual ghosts found in theories like Pauli-Fierz gravity. Further they obey certain nonrenormalization theorems, so that we are actually studying the exact object when you deal with quantum mechanics.
They also have interesting cosmological behavior, in that they can create accelerating spacetimes similar to the DGP setup and can thus potentially explain dark energy without the need for a cosmological constant.
