the interior product
The interior product is the operation of plugging a vector field into the first slot of a differential form. If a k-form is something waiting to eat k vectors, then feeding it one vector field X up front leaves a (k-1)-form still waiting for the rest. It lowers degree by one, the opposite of what the exterior derivative does. Intuitively it 'contracts' a form against a direction of flow.
Precisely, for a vector field X and a k-form omega, the interior product iota_X omega (also written i_X omega or X contracted into omega) is the (k-1)-form defined by (iota_X omega)(v_1, ..., v_{k-1}) = omega(X, v_1, ..., v_{k-1}). On 0-forms (functions) it gives zero. It is an antiderivation of degree -1: iota_X(alpha ^ beta) = (iota_X alpha) ^ beta + (-1)^{deg alpha} alpha ^ (iota_X beta), and it squares to zero, iota_X iota_X = 0. It is also linear over functions in the slot X, unlike d. In coordinates, iota_{partial/partial x^j} simply deletes any dx^j it finds, with the appropriate sign from moving it to the front.
The interior product is the third member, alongside d and the Lie derivative, of the trio bound together by Cartan's magic formula L_X = d iota_X + iota_X d. It is how you turn a top-degree volume form into a flux form (contracting the volume form against a vector field gives the form whose integral over a boundary computes flux, the geometric content of the divergence theorem), and it is essential in symplectic geometry, where iota_X omega = dH defines a Hamiltonian vector field. A note on names: 'interior product' has nothing to do with an inner product / dot product — no metric is involved; the unfortunate clash of names is purely historical.
On R^3 take the volume form mu = dx ^ dy ^ dz and the vector field X = (P, Q, R) = P partial_x + Q partial_y + R partial_z. Then iota_X mu = P dy^dz - Q dx^dz + R dx^dy = P dy^dz + Q dz^dx + R dx^dy, which is exactly the flux 2-form of the field X. Stokes' theorem applied to it is the divergence theorem. And iota_{partial_x}(dx^dy) = dy, while iota_{partial_y}(dx^dy) = -dx (the sign comes from sliding dy past dx).
Contracting the volume form against a vector field yields its flux form — this is how the divergence theorem becomes a special case of Stokes.
Interior product is not an inner product; it consumes a vector and a form, not two vectors, and needs no metric. Also it is C-infinity-linear in its vector argument (unlike d, which is only R-linear), so iota_{fX} omega = f iota_X omega for any function f — a useful check.