I find Wikipedia's discussion of symbols of differential operators a bit impenetrable, and Google doesn't seem to turn up useful links, so I'm hoping someone can point me to a more pedantic discussion.
Background
I think I understand the basic idea on ℝn, so for readers who know as little as I do, I will provide some ideas. Any differential operator on ℝn is (uniquely) of the form Σ pi1,...,ik(x) ∂k/(∂xi1...∂xik), where x1,...,xn are the canonical coordinate functions on ℝn, the pi1,...,ik(x) are smooth functions, and the sum ranges over (finitely many) possible indexes (of varying length). Then the symbol of such an operator is Σ pi1,...,ik(x) ξi1...ξik, where ξ1,...,ξn are new variables; the symbol is a polynomial in the variables {ξ1,...,ξn} with coefficients in the algebra of smooth functions on ℝn.
Ok, great. So symbols are well-defined for ℝn. But most spaces are not ℝn — most spaces are formed by gluing together copies of (open sets in) ℝn along smooth maps. So what happens to symbols under changes of coordinates? An affine change of coordinates is a map yj(x) = aj + Σi Yjixi for some vector (a1,...,an) and some invertible matrix Y. It's straightforward to describe how the differential operators change under such a transformation, and thus how their symbols transform. In fact, you can forget about the fact that indices range 1,...,n, and think of them as keeping track of tensor contraction; then everything transforms as tensors under affine coordinate changes, e.g. the variables ξi transform as coordinates on the cotangent bundle.
On the other hand, consider the operator D = ∂2/∂x2 on ℝ, with symbol ξ2; and consider the change of coordinates y = f(x). By the chain rule, the operator D transforms to (f'(y))2 ∂2/∂y2 + f''(y) ∂/∂y, with symbol (f'(y))2 ψ2 + f''(y) ψ. In particular, the symbol did not transform as a function on the cotangent space. Which is to say that I don't actually understand where the symbol of a differential operator lives in a coordinate-free way.
Why I care
One reason I care is because I'm interested in quantum mechanics. If the symbol of a differential operator on a space X were canonically a function on the cotangent space T*X, then the inverse of this Symbol map would determine a "quantization" of the functions on T*X, corresponding to the QP quantization of ℝn.
But the main reason I was thinking about this is from Lie algebras. I'd like to understand the following proof of the PBW theorem:
Let L be a Lie algebra over ℝ or ℂ, G a group integrating the Lie algebra, UL the universal enveloping algebra of L and SL the symmetric algebra of the vector space L. Then UL is naturally the space of left-invariant differential operators on G, and SL is naturally the space of symbols of left-invariant differential operators on G. Thus the map Symbol defines a canonical vector-space (and in fact coalgebra) isomorphism UL → SL.