the curvature form
Curvature is what stops a curved space from being flat, made local and infinitesimal. Take the holonomy idea — parallel transport around a loop returns a vector rotated — and shrink the loop to a point: the rotation per unit enclosed area, in the limit, is the curvature. It measures how badly the horizontal directions chosen by a connection fail to fit together into integrable slices. Where curvature vanishes the bundle is locally trivial-with-flat-connection; where it does not, no choice of local frame can make the connection disappear.
Precisely, for a connection 1-form omega on a principal bundle the curvature is the g-valued 2-form Omega given by the Cartan structure equation Omega = d omega + omega ^ omega (the wedge here combines the form wedge with the Lie bracket of g; for a matrix group it is literally matrix multiplication of forms, often written (1/2)[omega, omega]). In a local trivialization with gauge potential A this reads F = dA + A ^ A, the field strength of gauge theory; for an abelian group the quadratic term drops and F = dA. Unlike the connection, the curvature IS a tensor: under a gauge change h it transforms homogeneously, F -> h^{-1} F h, with no inhomogeneous derivative term. On a vector bundle the same object is the End(E)-valued 2-form R(X,Y) = nabla_X nabla_Y - nabla_Y nabla_X - nabla_{[X,Y]}, the failure of covariant derivatives to commute.
Curvature is the source of everything downstream. Chern-Weil theory feeds it into invariant polynomials to produce the characteristic classes; the Bianchi identity d Omega + [omega, Omega] = 0 constrains it; and in physics F is the electromagnetic or Yang-Mills field strength whose square is the action. Two honest cautions: first, curvature is a local, pointwise quantity, while holonomy is its global integral — do not conflate them. Second, sign and factor conventions (whether the quadratic term carries a 1/2, the sign of the bracket) differ between Kobayashi-Nomizu, Lee, and the physics literature, so always state which convention you are using before trusting a formula.
For a U(1) (abelian) connection, write A = A_mu dx^mu locally; then F = dA has components F_{mu nu} = partial_mu A_nu - partial_nu A_mu — exactly the electromagnetic field tensor, with the electric and magnetic fields as its entries. The Maxwell equation dF = 0 (one half of Maxwell) is just the Bianchi identity here, automatic because F = dA and d^2 = 0.
Electromagnetism is the curvature of a U(1) connection; its field strength F = dA is a curvature 2-form.
Curvature is a tensor (transforms homogeneously) whereas the connection is not — so 'curvature zero' is a coordinate-free, gauge-invariant statement (the connection is locally flat), but 'connection zero' is not. Also, curvature is pointwise; holonomy is its loop integral. And mind the conventions: the 1/2 on the quadratic term and the bracket sign vary by textbook.