an elliptic operator
Laplace's equation says that a quantity sits exactly at the average of its neighbours — perfect equilibrium with no preferred direction. An elliptic operator is the natural generalisation: it still describes equilibrium, but now the medium may be non-uniform and direction-dependent. Heat may flow more easily along the grain of a piece of wood than across it; an elliptic operator captures that anisotropy while keeping the rigid, smoothing, equilibrium flavour of the Laplacian.
A general second-order operator in n variables is L u = - sum over i,j of a_ij(x) u_(x_i x_j) + sum over i of b_i(x) u_(x_i) + c(x) u. The principal part — the highest-order piece — is the matrix a_ij(x) acting on the second derivatives. The operator is elliptic at a point x if that matrix a_ij(x) is positive definite there: for every nonzero direction vector xi, sum over i,j of a_ij(x) xi_i xi_j is strictly positive. Equivalently, the symbol sum a_ij xi_i xi_j never vanishes for xi not zero, so there are no real characteristic directions. The Laplacian itself has a_ij = the identity matrix, the simplest possible positive-definite case. The defining feature is geometric: the level sets of the symbol are ellipsoids, which is exactly where the name comes from.
Ellipticity is the algebraic signature of all the good behaviour of equilibrium equations: a solution is determined by its values on the whole boundary (not by initial data on one slice), solutions are smooth wherever the coefficients are, and the maximum principle holds. It is the opposite end of the spectrum from hyperbolic operators like the wave operator, which propagate signals along characteristics at finite speed. An elliptic operator has no characteristics to propagate along — information is felt everywhere at once.
On a sheet of plywood with conductivity twice as large along the grain (x) as across it (y), the steady temperature solves - (2 u_xx + u_yy) = 0. The principal matrix is diag(2, 1), positive definite, so the operator is elliptic; its symbol 2 xi_1^2 + xi_2^2 is positive for every nonzero (xi_1, xi_2), confirming there are no characteristic directions.
Anisotropy stretches the ellipse but does not destroy ellipticity — the symbol stays positive in every direction.
Ellipticity is a condition on the principal (top-order) part only; lower-order terms b_i and c never affect it. But those lower-order terms do matter for solvability — a large positive c helps, while a c that is too negative can let the operator hit an eigenvalue and lose uniqueness.