Quote:
Originally Posted by Damian
I know, I wanted a less formal definition, maybe with some numerical example, to make it less abstract and more easy to understand it.

I suggest you the very nice book by Nakahara: "Geometry, Topology and Physics" (if I recall correctly). It has lots of explicit examples etc. on this and related subjects (even if there are typos in several formulae, but usually just signs (+/) or so.)
PS: it seems there is some explicit calculation on
http://en.wikipedia.org/wiki/Connection_(mathematics)
PPS: well, not much... I think you have to plug in those into the formulae on the page "covariant derivative"
In fact, there are different notions of covariant derivatives. In general, "covariant" is w.r.t. some local ("gauge") transformation. In general relativity, there are 2 such transformations to be considered : local Lorentz transformations (SU(2) or SO(3,1) acting on "Lorenz" indices), and local coordinate transformations (diffeomorphisms ; acting on "Einstein indices"). "of course", both are linked...