a connection form
Move a vector along a curve on a curved space and ask: did it 'stay the same'? On a flat plane the answer is obvious — keep the components fixed. On a sphere it is not, because the tangent spaces at different points are different planes and there is no canonical way to compare them. A connection is the missing rule: it tells you how to identify nearby fibers, i.e. how to differentiate sections and how to transport them. There are two equivalent ways to package this rule — as a splitting of directions (horizontal versus vertical), and as a 1-form valued in the symmetry algebra.
On a principal G-bundle P, the vertical directions at a point are those tangent to the fiber (moving within G), and a connection chooses a complementary horizontal subspace at every point, varying smoothly and G-invariantly. Equivalently, a connection is a 1-form omega on P valued in the Lie algebra g that (i) reproduces the generator of the group action on vertical vectors and (ii) transforms by the adjoint representation under the right G-action. The horizontal space is then ker(omega), and 'horizontal' means 'moving without rotating the fiber'. In a local trivialization omega pulls back to a g-valued 1-form A on the base — the gauge potential of physics — and changing trivialization by h: U -> G transforms it by the inhomogeneous law A -> h^{-1} A h + h^{-1} dh, the dh term being what distinguishes a connection from an ordinary tensor.
The connection form is the central object of bundle geometry: it defines parallel transport (lift a curve horizontally), the covariant derivative (differentiate while subtracting the connection's correction), curvature (the failure of horizontal subspaces to integrate), and holonomy (the net transport around loops). One honest subtlety: a connection is not a tensor — its transformation law has the extra h^{-1} dh term, so 'the connection is zero' is a statement that depends on the trivialization, and is meaningful only locally. Differences of connections, however, are tensors (g-valued 1-forms on the base), so connections form an affine space.
On the trivial bundle R^2 times R^k take A = 0: parallel transport is 'keep components constant', the flat connection. Now on the orthonormal frame bundle of S^2 with the round metric, the Levi-Civita connection has a nonzero connection form; transporting a vector around a spherical triangle of area Area rotates it by exactly Area (the angle deficit). The same vector returns rotated, even though it was 'never turned' relative to the connection — that rotation is holonomy, produced by the connection's curvature.
A connection is a transport rule; on a curved space it returns vectors rotated, and that rotation is the visible face of curvature.
A connection is not a tensor: its gauge-transformation law A -> h^{-1}Ah + h^{-1}dh has the inhomogeneous h^{-1}dh term, so you can always make A vanish at a single point by choosing a frame, but you cannot in general make it vanish in a neighborhood — that would force the curvature to be zero. (Differences of connections, by contrast, are genuine tensors.)