the principal symbol
If you want to know how a car will handle at speed, you study its engine and chassis, not the cup-holders. For a differential operator, the part that governs its essential character — its type, its smoothness behaviour, how disturbances propagate — is the highest-order part. The principal symbol is what you get by feeding ONLY those top-order terms through the symbol dictionary, throwing away the low-order clutter. It is the engine of the operator made visible.
Start from the full symbol (replace each derivative D_j by i*xi_j) and keep only the terms of the maximal differential order. For a second-order operator that is the homogeneous degree-2 polynomial in xi coming from the principal part A u_xx + 2B u_xy + C u_yy, namely the quadratic form -(A xi_1^2 + 2B xi_1 xi_2 + C xi_2^2). All the first-order and zeroth-order terms are dropped. The point of keeping only this piece is that the type is decided entirely by it: the equation is elliptic if this quadratic form never vanishes for xi not equal to 0 (it is definite), parabolic if it is degenerate (some non-zero xi makes it vanish but it does not change sign), and hyperbolic if it is indefinite (it takes both signs). In two variables that test reproduces exactly the discriminant rule B^2 - AC.
The principal symbol is the right tool because, unlike the lower-order terms, it transforms cleanly under a change of variables (it behaves like a well-defined function on the cotangent space) and it is what survives in every asymptotic, high-frequency limit. It generalizes the two-variable discriminant to any number of dimensions and underlies elliptic regularity, the propagation of singularities along the characteristics (the zero set of the principal symbol), and the entire calculus of pseudodifferential and Fourier-integral operators. The honest caveat: the lower-order terms it ignores are not irrelevant to actually solving the equation — they can change uniqueness, decay, and the fine details — but they never change the type.
The operator L = u_xx + u_yy + 7 u_x - 3u has full symbol -(xi_1^2 + xi_2^2) + 7 i xi_1 - 3, but its principal symbol is just -(xi_1^2 + xi_2^2). Since that never vanishes unless xi = 0, L is elliptic — and the 7 u_x and -3u terms, however they affect the solution's details, cannot change that verdict.
Keep only the top-order part: the principal symbol decides the type, and the lower-order terms cannot overturn it.
The principal symbol, not the full symbol, is the classifier — and it is precisely the lower-order terms it discards that the type is insensitive to. Their zero set (where the principal symbol vanishes) is the characteristic set, along which singularities travel.