Non-Affine Connections: Why & What?

[Geometry_dude]

"Everyone" knows what an affine connection on a smooth manifold is a.k.a. covariant derivative. My questions are:
i) Why are those connections called affine?
ii) Is there a mathematical object that 'connects neighboring tangent spaces', that could be termed a 'non-affine connection'?

 

[Matterwave]

I quote wikipedia: "The terminology [affine connection] is due to Cartan and has its origins in the identification of tangent spaces in Euclidean space Rⁿ by translation: the idea is that a choice of affine connection makes a manifold look infinitesimally like Euclidean space not just smoothly, but as an affine space."

http://en.wikipedia.org/wiki/Affine_connection