Hilbert space
A Hilbert space is the setting where the full geometry of Euclidean space — lengths, angles, perpendicularity, the Pythagorean theorem — survives intact even in infinitely many dimensions. The extra ingredient beyond a Banach space is an inner product, which lets you ask not only how long a vector is but how two vectors are oriented relative to each other. It is the natural home of Fourier analysis and quantum mechanics.
Precisely, a Hilbert space is an inner product space (a vector space with an inner product <x, y>) that is complete with respect to the norm ||x|| = sqrt(<x, x>) it induces. Completeness here means the same as for Banach spaces: every Cauchy sequence converges inside the space. So a Hilbert space is exactly a Banach space whose norm comes from an inner product.
Not every norm arises from an inner product; the ones that do are characterized by the parallelogram law ||x + y||^2 + ||x - y||^2 = 2||x||^2 + 2||y||^2. When it holds, the inner product can be recovered from the norm by polarization, and the rich apparatus of orthogonality unlocks: orthonormal bases, orthogonal projection onto closed subspaces, and the Riesz representation of every continuous functional. The canonical example is l^2, square-summable sequences, the infinite-dimensional analogue of R^n.
The space L^2[-pi, pi] with inner product <f, g> = integral f(x) conjugate(g(x)) dx is a Hilbert space. The functions e_n(x) = exp(i n x)/sqrt(2 pi) form an orthonormal basis, and expanding f in this basis is exactly its Fourier series — geometry in infinite dimensions made into a computational tool.
L^2 with the complex exponentials: a Hilbert space whose orthonormal basis is the Fourier series.
All separable infinite-dimensional Hilbert spaces are isometrically isomorphic to l^2 — there is essentially only one. This rigidity is special to Hilbert spaces; Banach spaces come in a vast and wild variety with no such classification.