TITLE: Covariant derivative geometric interpretation
QUESTION [7 upvotes]: I'm having some trouble understanding what the covariant derivative means geometrically. I know the definition which states that for a tensor T with any number of indices: $ \nabla_j T = \frac{\partial T}{\partial Z^k} + $ the Christoffel symbols contracted with each index of the tensor appropriately. 
In the case of an invariant, I can visualize this as the normal partial derivative, so the geometric interpretation is clear. Also, for a first-order tensor $V^i$ such that vector $ \textbf{R} = V^i \textbf{Z}_i$, I know that I can say $$ \frac{\partial \textbf{R}}{\partial Z^k} = \nabla_k (V^i) \textbf{Z}_i $$ However, how can I understand the covariant derivative of higher order tensors and of vectors? What is the geometric meaning?

REPLY [3 votes]: @mollyerin gives a nice answer. Furthermore, depending on the rigor that you are looking for I think that Roger Penrose's book Road to Reality gives a nice overview of the geometric interpretation of covariant derivatives. It should be noted that the book attempts to present topics such as the covariant derivative and other mathematical subjects using layman's terms.