The principal symbol of a differential operator $\sum__ {|\alpha| \leq m} a__ \alpha(x) \partial__ x^\alpha$ is by definition the function $\sum__ {|\alpha| = m} a__ \alpha(x) (i\xi)^\alpha$ Here \alpha is a multi-index (so \partial__ x^\alpha denotes \alpha__ 1 derivatives with respect to x__ 1, etc.) At this point, the vector \xi = (\xi__ 1, \ldots, \xi__ n) is merely a formal variable. The power of this definition is that if one interprets (x,\xi) as variables in the cotangent bundle in the usual way -- i.e. x is any local coordinate chart, then \xi is the linear coordinate in each tangent space using the basis dx^1, \ldots, dx^n, then the principal symbol is an invariantly defined function on T^*X, where X is the manifold on which the operator is initially defined, which is homogeneous of degree m in the cotangent variables.
Here is a more invariant way of defining it: fix (x__ 0,\xi__ 0) to be any point in T^*X and choose a function \phi(x) so that d\phi(x__ 0) = \xi__ 0. If L is the differential operator, then L( e^{i\lambda \phi}) is sum complicated sum of derivatives of \phi, multiplied together, but always with a common factor of e^{i\lambda \phi}. The `top order part' is the one which has a \lambda^m, and if we take only this, then its coefficient has only first derivatives of \phi (lower order powers of \lambda can be multiplied by higher derivatives of \phi). Hence if we take the limit as \lambda \to \infty of \lambda^{-m} L( e^{i\lambda \phi}) and evaluate at x = x__ 0, we get something which turns out to be exactly the principal symbol of L at the point (x__ 0, \xi__ 0).
There are many reasons the principal symbol is useful. There is indeed a `quantization map' which takes a principal symbol to any operator of the correct order which has this as its principal symbol. This is not well defined, but is if we mod out by operators of one order lower. Hence the comment in a previous reply about this being an isomorphism between filtered algebras.
In special situations, e.g. on a Riemannian manifold where one has preferred coordinate charts (Riemann normal coordinates), one can define a total symbol in an invariant fashion (albeit depending on the metric). There are also other ways to take the symbol, e.g. corresponding to the Weyl quantization, but that's another story.
In microlocal analysis, the symbol captures some very strong properties of the operator L. For example, L is called elliptic if and only if the symbol is invertible (whenever \xi \neq 0). We can even talk about the operator being elliptic in certain directions if the principal symbol is nonvanishing in an open cone (in the \xi variables) about those directions. Another interesting story is wave propagation: the characteristic set of the operator is the set of (x,\xi) where the principal symbol p(L) vanishes. If its differential (as a function on the cotangent bundle) is nonvanishing there, then the integral curves of the Hamiltonian flow associated to p(L), i.e. for the Hamiltonian vector field determined by p(L) using the standard symplectic structure on T^*X, ``carries'' the singularities of solutions of Lu = 0. This is the generalization of the classical `fact' that singularities of solutions of the wave equation propagate along light rays.