(Since the manifold metric is always assumed to be regular, the compatibility condition implies linear independence of the partial derivative tangent vectors.)
and yields the Christoffel symbols for the Levi-Civita connection in terms of the metric:
For a very simple example that captures the essence of the description above, draw a circle on a flat sheet of paper. Travel around the circle at a constant speed. The derivative of your velocity, your acceleration vector, always points radially inward. Roll this sheet of paper into a cylinder. Now the (Euclidean) derivative of your velocity has a component that sometimes points inward toward the axis of the cylinder depending on whether you're near a solstice or an equinox. (At the point of the circle when you are moving parallel to the axis, there is no inward acceleration. Conversely, at a point (1/4 of a circle later) when the velocity is along the cylinder's bend, the inward acceleration is maximum.) This is the (Euclidean) normal component. The covariant derivative component is the component parallel to the cylinder's surface, and is the same as that before you rolled the sheet into a cylinder.
The first term in this formula is responsible for "twisting" the coordinate system with respect to the covariant derivative and the second for changes of components of the vector field u. In particular
In words: the covariant derivative is the usual derivative along the coordinates with correction terms which tell how the coordinates change.
where semicolon ";" indicates covariant differentiation and comma "," indicates partial differentiation. Incidentally, this particular expression is equal to zero, because the covariant derivative of a function solely of the metric is always zero.
In textbooks on physics, the covariant derivative is sometimes simply stated in terms of its components in this equation.
In some older texts (notably Adler, Bazin & Schiffer, Introduction to General Relativity), the covariant derivative is denoted by a double pipe and the partial derivative by single pipe:
The derivative along a curve is also used to define the parallel transport along the curve.
A covariant derivative introduces an extra geometric structure on a manifold that allows vectors in neighboring tangent spaces to be compared: there is no canonical way to compare vectors from different tangent spaces because there is no canonical coordinate system.