Picturing the how the covariant derivative acts on vector fields

Question

I am curious how people picture geometrically how the covariant derivative acts with regards to vector fields. I understand that it is merely an object which has been defined for some use and obeys certain conditions but how would you picture such a thing geometrically? In other words, when someone asks you what the covariant derivative is, how do you explain it in absence of a formal definition?

Answer 1

Suppose that the manifold $M$ is embedded in $\mathbb R^n$. Then a vector $v \in T_p M$ at point $p \in M$ can be considered a vector in $\mathbb R^n$.

The covariant derivative $\nabla_u v$ for some $u \in T_p M$ then is the ordinary derivative of $v$ in the direction $u$ projected on $T_p M$.