One way of looking at General Relativity (one of whose core principles is independence of any fixed coordinate system), is to see it as a scalar field in five dimensional space (you could think of the fifth dimension in General Relativity as a point's "relativistic character") rather than as a tensor field in four dimensional place, as it is conventionally described. Analysis of the symmetries of this five dimensional scalar field representation of General Relativity suggested the idea of a new kind of symmetry that can exist in any kind of scalar field called a Galilean symmetry. As the Cosmic Variance blog explains:
[A scalar field that obeys the Galilieon symmetry is a] scalar field theory with a single complicated derivative interaction, obeying the galileon symmetry under which the action is invariant when derivatives of the field are shifted by a constant vector. . . .
[T]he realization that there existed a previously unconsidered symmetry of scalar field theories led Nicolis, Rattazzi and Trincherini to consider abstracting the symmetry, and asking what other terms may be allowed for scalar fields. And, remarkably, there turn out to only be five! In this abstracted scalar field theory we refer to these terms as the galileon terms, and to the scalar field itself as the Galileon.
[T]here are a number of properties they have that should illustrate why a number of people in the community have found them sufficiently interesting to warrant further study.
1. The Galileon terms involve higher derivatives, but their equations of motion are only second order in time, and hence they avoid some well-known proofs of instability that plague a lot of higher derivative systems.
2. There exists a range of energy scales over which the Galileon terms are important, and hence higher derivatives are important, yet quantum mechanical effects are irrelevant, and classical physics holds.
3. The Galileon terms are unrenormalized! Their coefficients pick up no modifications from quantum corrections arising from other Galileon terms!
This last feature hints at a number of possible applications in cosmology. For example, cosmic acceleration, either in the early or the late universe, typically requires scalar fields with dynamics that are finely tuned, and hence are easily perturbed by quantum corrections. There is therefore the possibility that Galileons may lead to a natural way to achieve such behavior.
In other words, if you take the principle of coordinate system independence, and bound by that assumption project a mathematically equivalent version of a complicated theory like general relativity into more dimensions, you can end up with simpler equations that have to fit a very constrained form.
By analogy, the point, line, circle, ellipse, parabola, and hyperbola in two dimensions can all be described as slices of a pair of cones that meet at the tip and share a common axis (i.e. as conic sections), that can be described by a very narrow set of polynomial equations with a small number of generalizable properties.
Any time that we know that a law of physics must follow Galileon symmetries, we can hope to considerably narrow the range of equations that could possibly describe it from the universe of equations that would be eligible if we didn't know that this constraint existed. And, we can also better focus the efforts of our mathematical theorists if they only have to figure out how to solve equations that can only have one or more of five very specific kinds of terms, than we could if we had to solve more general versions of the same equations.
Knowing the limited kinds of terms that Galileon scalar field equations can take also makes it very easy to immediately determine that an equation proposed to describe a scalar field with a Galileon symmetry is incorrect because it has the wrong kind of term in it without doing any other analysis of what those equations imply for the real world.
Since one of the main problems in developing the math necessary for a quantum theory of gravity is that it is non-renormalizable, but you can create a Galileon set of equations for general relativity for which in inability to renormalize is not a problem, this mathematical trick has the potential to be a big deal for quantum gravity theorists.
This is particularly useful in considering the idea that gravitons have mass, or alternately that the energy of gravitational fields themselves contribute to the mass-energy tensor, and what that implies in terms of modifications of gravitational theories that could have an impact on our understanding of dark matter and dark energy.
A 2011 paper, illustrates this concept in a context easier to grasp than the cosmological inflation example cited at Cosmic Variance is that it helps explain how dark energy could function in very weak gravitational fields (such as those found in deep space between galaxies) but not in strong ones such as the gravitational field we observe in the solar system.
Spelling note: The term Galileon is derived from the name of the scientist Galieo. I have seen both "Galileon" and "Galilean" used in print for this term, for example, even in the same post at Cosmic Variance. I have perferred the former, given its origins, and have imposed consistency of spelling on this post. It may be that different spellings are used in different senses of the word and that I have not read carefully enough to discern them.