a coadjoint orbit
A Lie group acts on its own Lie algebra by the adjoint action (rotating the algebra by conjugation), and dually it acts on the dual of the Lie algebra by the coadjoint action. The orbits of this dual action — the sets you reach from a given functional by applying all group elements — are the coadjoint orbits. The striking fact is that every single one of these orbits is automatically a symplectic manifold, with a canonical symplectic form requiring no extra choices. They are the most natural family of symplectic manifolds in all of mathematics.
Precisely, let G be a Lie group with Lie algebra g and dual g^*. The coadjoint action of G on g^* is the dual of the adjoint action: g . xi for g in G, xi in g^*. The orbit O_xi = {Ad^*_g xi : g in G} carries the Kirillov-Kostant-Souriau (KKS) symplectic form, defined at a point xi by omega_xi(ad^*_X xi, ad^*_Y xi) = <xi, [X, Y]>, the pairing of xi with the Lie bracket. This formula is well-defined and nondegenerate precisely on the tangent space to the orbit, and the Jacobi identity for [.,.] makes it closed — so (O_xi, omega) is a symplectic manifold with no choices made. Coadjoint orbits are exactly the symplectic leaves of the natural Lie-Poisson structure on g^*.
Why they matter: coadjoint orbits are the symplectic-geometry side of representation theory. Kirillov's orbit method proposes that irreducible unitary representations of G correspond to (quantizations of) coadjoint orbits — for nilpotent and many solvable and compact groups this is a precise dictionary. They are also the model fibers of symplectic reduction (every coadjoint orbit is a reduction of T*G), and for compact groups they are coadjoint orbits = flag manifolds, simultaneously Kähler and symplectic. A point to keep straight: the canonical symplectic form lives on each orbit individually, not on g^* itself — g^* is only a Poisson manifold, foliated by the symplectic coadjoint orbits, and orbits of different 'sizes' (e.g. through regular versus singular elements) have different dimensions.
For G = SO(3), the dual so(3)^* is R^3 and the coadjoint action is ordinary rotation. The orbits are the concentric spheres of radius r > 0 (plus the single fixed point at the origin). The KKS form on a sphere of radius r is r times the standard area form, so each sphere S^2_r is symplectic with total area 4 pi r^2. Quantizing these (Bohr-Sommerfeld: area an integer multiple of 2 pi hbar) recovers exactly the spin-j representations of SU(2), the orbit method in its cleanest form.
For SO(3), coadjoint orbits are spheres in R^3; quantizing them gives the spin representations.
The symplectic structure lives on each orbit, not on g^* — g^* is a Poisson manifold whose symplectic leaves are the coadjoint orbits. Different orbits generally have different dimensions, and the orbit through 0 is a single point, not a positive-dimensional symplectic manifold.