a symplectic manifold
Think of the state of a moving particle: you need both where it is (position) and how it is moving (momentum). Bundle these together and you get phase space, the natural arena of classical mechanics. A symplectic manifold is the abstract geometry of such a phase space. Where a Riemannian manifold measures lengths and angles with a metric, a symplectic manifold instead measures oriented areas of little parallelograms with a special 2-form, and it is this area-measuring structure — not distance — that drives Hamilton's equations of motion.
Precisely, a symplectic manifold is a pair (M, omega) where M is a smooth even-dimensional manifold and omega is a differential 2-form that is closed (d omega = 0) and nondegenerate (at each point, if omega(v, w) = 0 for all w then v = 0). Nondegeneracy forces the dimension to be even, say 2n, and makes omega^n a nowhere-vanishing top form, so M is automatically orientable with a canonical volume form omega^n / n!. The closedness d omega = 0 is the deep condition: it is what lets omega^n behave like a conserved volume and what makes the local model rigid. The basic example is R^{2n} with coordinates (q_1, ..., q_n, p_1, ..., p_n) and omega = sum dp_i ^ dq_i; the basic geometric example is any cotangent bundle T*N with its canonical 2-form.
Why it matters: symplectic geometry is the coordinate-free language of Hamiltonian mechanics, and through Darboux's theorem it has a startling feature — there are no local invariants. Two symplectic manifolds of the same dimension look identical in small enough patches, unlike Riemannian manifolds, which curvature distinguishes locally. So there is no 'symplectic curvature': all the geometry is global and topological. A common confusion is to imagine omega as a kind of metric; it is not. It is antisymmetric (omega(v, w) = -omega(w, v)), so omega(v, v) = 0 — a vector has zero 'symplectic self-pairing', which is exactly why area, not length, is the natural measurement.
On R^2 with coordinates (q, p) take omega = dp ^ dq. For two vectors v = (a, b) and w = (c, d), omega(v, w) = ad - bc, which is exactly the signed area of the parallelogram they span. This is nondegenerate (only the zero vector pairs to zero with everything) and closed (d omega = 0 since omega is constant-coefficient), so (R^2, dp ^ dq) is the simplest symplectic manifold — the phase plane of a single one-dimensional particle.
The phase plane (R^2, dp ^ dq): omega measures signed area of parallelograms, not length.
Not every even-dimensional manifold is symplectic. On a closed manifold, [omega] must be a nonzero class in H^2(M; R) with [omega]^n nonzero, which already rules out, e.g., S^4 and S^{2n} for n > 1. Symplectic existence is a genuine topological constraint, even though there are no local obstructions.