quantum mechanics - Electromagnetic Field as a Connection in a Vector Bundle

I would like to know more about Ehresmann connections in vector bundles and how they relate to the electromagnetic field and the electron in quantum mechanics.

Background: The Schrödinger equation for a free electron is

$$\frac{(-i\hbar\nabla)^2}{2m} \psi = i\hbar\partial_t \psi$$

Now, to write down the Schrödinger equation for an electron in an electromagnetic field given by the vector potential $A=(c\phi,\mathbf{A})$, we simply replace the momentum and time operator with the following operators

$$\begin{array}{rcl} -i\hbar\nabla &\mapsto& D_i = -i\hbar\nabla + e\mathbf{A} \\ i\hbar\partial_t &\mapsto& D_0 = i\hbar\partial_t - e\phi \end{array}$$

I have heard that this represents a "covariant derivative", and I would like to know more about that.

My questions:

1. 
   

   (Delegated to Notation for Sections of Vector Bundles.)

   
   


2. 
   

   I have heard that a connection is a "Lie-algebra-valued one-form". How can I visualize that? Why does it take values in the Lie-algebra of $U(1)$?

   
   


3. 
   
   

   Since a connection is a one-form, how can I apply it to a section $\psi$? I mean, a one-form eats vectors, but I have a section here? What is $D_\mu \psi(x^\mu)$, is it a section, too?

   
   

I apologize for my apparent confusion, which is of course the reason for my questions.