Before this page I already had these pages of the same title: one, two and three. I must consolidate those but meanwhile:
There is a vital aspect of affine connection not yet captured in my other pages. An affine connection on a manifold defines the meaning of parallel motion of a vector along a definite path in the manifold. (Actually same direction and magnitude—same vector.)
An unnamed class of affine transformations have the property of maintaining the determinant of the transform between any two points. When the path is a loop, the determinant of the resulting transformation is 1. Perhaps these should be called unimodular affine connections. The affine connection from a metric is unimodular.
Shild’s ladder provides an intuitive construction of parallel transport given the notion of geodesics, which in turn stems from a metric.
Two PlanetMath pages: Connections and affine connection.