the symbol of an operator

Differentiation is a slippery thing to compute with directly, but there is a magic dictionary that turns it into ordinary algebra. The Fourier transform sends a derivative into multiplication: differentiating once becomes multiplying by i*xi (where xi is the frequency variable). Apply that dictionary to a whole differential operator and out pops a polynomial — its symbol. The symbol is the operator rewritten as plain algebra, and reading it is how you classify the operator without ever solving anything.

Take a linear differential operator, say L = sum of a_alpha(x) D^alpha (a combination of partial derivatives of various orders with coefficients). Its symbol is obtained by the substitution rule: replace each derivative D_j with i*xi_j, so the operator becomes a polynomial in the frequency vector xi = (xi_1, ..., xi_n). For example the Laplacian Delta = u_xx + u_yy becomes (i*xi_1)^2 + (i*xi_2)^2 = -(xi_1^2 + xi_2^2); the heat operator d/dt - Delta becomes i*tau + (xi_1^2 + ...). This rule works because e^{i x . xi} is an eigenfunction of every constant-coefficient differential operator: applying the operator just multiplies it by the symbol. The symbol thus tells you exactly how the operator acts on each frequency.

Why this matters: the symbol converts every question about the differential operator into a question about a polynomial. The operator's type, its invertibility, the smoothness of its solutions, and how it acts on waves of each frequency are all read off the symbol — which is the doorway to Fourier analysis, fundamental solutions, and the whole modern (pseudodifferential) theory. The honest fine print: for variable coefficients the symbol depends on the point x as well as the frequency xi, and the clean Fourier picture is exact only for constant coefficients (it becomes a local approximation otherwise). For classification, what actually matters is the top-degree part, the principal symbol.