the covariant derivative on a bundle
How do you differentiate a section of a vector bundle? You want to ask how a field of vectors changes as you move, but the vectors at different points live in different fibers, so the naive difference quotient subtracts things in different vector spaces — nonsense. The covariant derivative repairs this: it is a rule nabla that differentiates a section s in a direction X and produces another section nabla_X s, building in a correction so the answer is well defined and tensorial in X.
Precisely, a covariant derivative (or Koszul connection) on a vector bundle E -> M is a map nabla taking a vector field X and a section s to a section nabla_X s, that is linear and tensorial in X, additive in s, and obeys the Leibniz rule nabla_X (f s) = (X f) s + f nabla_X s for functions f. In a local frame e_1, ..., e_k it is recorded by connection coefficients (a matrix of 1-forms, the omega^a_b) via nabla e_b = omega^a_b e_a; in coordinates this is the same data as Christoffel-type symbols. The recipe in practice: to differentiate s = s^a e_a, write nabla_X s = (X(s^a) + omega^a_b(X) s^b) e_a — the ordinary derivative of the components plus the connection's correction that accounts for the frame turning. This is exactly the same connection as a connection 1-form on the associated frame bundle, viewed from the vector-bundle side.
The covariant derivative is what makes geometry on a bundle possible: a section is parallel along a curve when its covariant derivative along the curve vanishes (this is parallel transport), a geodesic is a curve whose velocity is parallel, and the failure of mixed second covariant derivatives to commute is the curvature, nabla_X nabla_Y - nabla_Y nabla_X - nabla_{[X,Y]} = R(X,Y). One caution worth stating: there is no canonical covariant derivative on a bare vector bundle — you must choose one, and the space of choices is affine (any two differ by an End(E)-valued 1-form). The Levi-Civita connection is the special canonical choice on the tangent bundle of a Riemannian manifold, and belongs to Riemannian geometry, not here.
On R^n the trivial connection is plain partial differentiation: nabla_X s = X(s^a) e_a, no correction. Now restrict the constant vector field 'pointing north' to a curve on the sphere and use the induced (Levi-Civita) connection: its covariant derivative along the curve is generally nonzero, because keeping a vector tangent to the sphere forces it to turn relative to the ambient frame. The component of the ambient derivative that stays tangent is exactly nabla; the part that sticks out of the surface is the second fundamental form.
Covariant derivative = ordinary derivative minus the part that leaves the bundle; on a surface that 'leaving part' is the second fundamental form.
There is no canonical covariant derivative on a general vector bundle — you must impose one, and any two differ by an End(E)-valued 1-form. The familiar uniqueness ('the connection') comes only on a Riemannian tangent bundle, where metric-compatibility plus torsion-freeness single out Levi-Civita; do not expect that uniqueness on an arbitrary bundle.