orthogonality with respect to a weight
Two arrows are perpendicular when their dot product is zero. Two functions are 'orthogonal' in the same spirit when a certain integral of their product is zero — they have no overlap, no shared component. Orthogonality with respect to a weight is this idea with the inner product measured using the weight w(x) of the problem.
Precisely, functions f and g are orthogonal with respect to the weight w on [a, b] when integral from a to b of f(x) g(x) w(x) dx = 0. The Sturm-Liouville theorem guarantees that eigenfunctions belonging to different eigenvalues are automatically orthogonal in exactly this weighted sense. The proof is a one-line consequence of self-adjointness: (lambda_m - lambda_n) times the integral of u_m u_n w equals a boundary term that vanishes, so when the eigenvalues differ the integral must be zero.
This is the engine that makes eigenfunction expansions practical. To find the coefficient c_n in f = sum c_n u_n, you multiply both sides by u_n w and integrate; every cross term dies by orthogonality, leaving c_n = (integral of f u_n w) / (integral of u_n^2 w). That is the generalized Euler-Fourier coefficient formula. Without weighted orthogonality you would face an infinite coupled system; with it, every coefficient is read off independently, exactly like projecting a vector onto perpendicular axes.
Bessel functions J_0(alpha_m x) and J_0(alpha_n x), where alpha_m are the zeros of J_0, are orthogonal on [0, 1] with weight w(x) = x: integral from 0 to 1 of J_0(alpha_m x) J_0(alpha_n x) x dx = 0 for m not n. To expand f(x) on a disk you use exactly this weighted integral to extract each coefficient.
Weighted orthogonality lets each expansion coefficient be computed by a single projection integral.
Eigenfunctions sharing the same eigenvalue (a degenerate eigenvalue) are not automatically orthogonal — you orthogonalize them by hand (Gram-Schmidt). The automatic orthogonality only applies across distinct eigenvalues.