In the past I had the feeling that I understood the mathematics and ideas behind principal connections and connection one-forms rather well. However, while trying to explain these ideas in simple terms to a nonmathematical audience, I noticed that my understanding might not extend flawlessly beyond the pure formulas.
When you have a principal $G$-bundle $\pi:P\rightarrow M$, you can canonically define the vertical subbundle as the kernel of the projection $\pi_*$. A connection is then equivalent to a choice of complement inside the tangent bundle $TP$. Locally, where $P|_U\cong U\times G$, one can identify the vertical spaces with the Lie algebra $\mathfrak{g}$. A complement can then be defined by assigning to every basis $\partial_i$ of a tangent space on $M$ a ''horizontal lift'' $$\widetilde{\partial}_i:=\partial_i+\chi_i,$$ where $\chi_i\in\mathfrak{g}$ (these two terms should be mapped in the right way to a tangent space of $P|_U$ and extended to a local frame).
The associated connection one-form $\omega$ on $P$ is then, as far as I understand, the form that assigns to any vector field on $P$, at any point, the contribution in $\mathfrak{g}$ that does not arise from these $\chi_i$'s: $$X = \sum_i\lambda_i(\partial_i+\chi_i) + \omega(X),$$ for some scalars numbers $\lambda_i$. It measures the change in the fibres that does not arise from a mere change on the base manifold.
When we then, locally, pullback the connection one-form $\omega$ to a one-form on the patch $U$, we get $$A = s^*\omega.$$ By definition of the pullback I would then think that this evaluates to the following formula on any vector field: $A(Y) = \omega(s_*Y)$. By then combining the above statements, I would expect that $A$ assigns to any vector field on $M$ the difference between its pushforward (along a section) to the bundle and its horizontal lift, resulting in a linear combination of the $\chi_i$'s. Is this correct?