a Laplacian eigenvalue
Strike a drumhead and it rings at certain pure pitches — its resonant frequencies. Mathematically each pure tone is an eigenfunction of the Laplacian, a vibration pattern that keeps its shape and only scales in amplitude, and its pitch-squared is the eigenvalue. The collection of all eigenvalues is the spectrum of the manifold, and it is the central object of spectral geometry: a numerical fingerprint that encodes (much of) the shape of the space.
On a compact Riemannian manifold (M, g) the Laplace-Beltrami operator has a discrete spectrum: there are eigenvalues 0 = lambda_0 < lambda_1 <= lambda_2 <= ... going to infinity, with smooth eigenfunctions phi_k solving Delta phi_k = lambda_k phi_k (using the geometer's sign so the spectrum is nonnegative). The eigenfunctions form an orthonormal basis of L^2(M), so every function expands in them — this is Fourier analysis adapted to the manifold. The lowest eigenvalue lambda_0 = 0 corresponds to constant functions, and on a connected manifold it is simple; the first nonzero eigenvalue lambda_1 measures how 'spread out' or 'connected' the manifold is and is the one most studied. Geometry controls these numbers through lower bounds: the Lichnerowicz estimate says that if Ricci curvature is bounded below by (n-1)k > 0 then lambda_1 >= n k (sharp on the round sphere), and the Cheeger inequality bounds lambda_1 below by (Cheeger constant)^2 / 4, tying the spectrum to how hard the manifold is to cut in two.
Laplacian eigenvalues govern heat diffusion (the heat kernel is sum e^{-lambda_k t} phi_k phi_k), wave propagation, and quantum energy levels on curved space, and their large-eigenvalue asymptotics (Weyl's law: the number of eigenvalues below lambda grows like (volume) lambda^{n/2}) recover dimension and volume. The honest caveats: the spectrum determines a lot but not everything. Two non-isometric manifolds can have identical spectra — they are isospectral but not isometric — so you cannot always 'hear the shape of a drum'. And eigenvalue bounds are usually one-sided and convention-sensitive; an upper bound for lambda_1 (like Cheeger's other inequality or Buser's) requires extra hypotheses, and the precise constants depend on the sign convention for Delta and on whether you index eigenvalues with or without multiplicity.
On the round sphere S^n of radius one, the first nonzero eigenvalue is lambda_1 = n, with eigenfunctions the restrictions of linear coordinate functions; this exactly saturates the Lichnerowicz bound lambda_1 >= n, since the sphere has Ricci curvature (n-1).
The round sphere is the equality case of the Lichnerowicz eigenvalue bound.
Isospectral but non-isometric manifolds exist, so the spectrum is a powerful but incomplete invariant — equal spectra do not imply the same shape, the precise content of the 'hearing the shape of a drum' story.