Let $\pi \colon P \rightarrow M$ be a principal $G$-bundle.
- Assume that we start with a notion of parallel transport. That is, given a smooth curve $\alpha \colon [0,1] \rightarrow M$ with $\alpha(0) = m$, we have the parallel transport maps $P_{\alpha, 0, t} \colon P_m \rightarrow P_{\alpha(t)}$ which allows us to identify the fiber $P_m = P_{\alpha(0)}$ over the starting point $m = \alpha(0)$ with the fibers $P_{\alpha(t)}$ along $\alpha$. Given a specific element $\sigma \in P_m$ in the fiber over the starting point, we can define a path $\tilde{\alpha}_{\sigma} \colon [0,1] \rightarrow P$ by $\tilde{\alpha}_{\sigma}(t) := P_{\alpha,0,t}(\sigma)$ (the parallel transport of $\sigma$ along $\alpha$). The path $\tilde{\alpha}_{\sigma}$ provides a lift of the path $\alpha$ with $\tilde{\alpha}_{\sigma}(0) = \sigma$. Given $m \in M$ and $\sigma \in P_m$, we can now define a subset $H_{(m,\sigma)}P \subseteq T_{(m,\sigma)}P$ called the horizontal space by
$$ H_{(m,\sigma)}P := \left \{ \frac{d}{dt} \tilde{\alpha}_{\sigma}|_{t = 0} \, : \, \alpha \in C^{\infty}([0,1],M), \alpha(0) = m \right \}.$$
The parallel transport maps are imposed to satisfy conditions which turns $HP$ into a smooth constant rank $G$-equivariant subbundle of $TP$ which complements the vertical bundle $VP = \ker(d\pi)$. For example, we want the lifts $\tilde{\alpha}_{\sigma}$ to be smooth so we require that the parallel transport maps are diffeomorphisms. We want the bundle to be $G$-equivariant so we require that the parallel transport maps are $G$-equivariant. We want $H_{(m,\sigma)}P$ to "depend smoothly" on $(m,\sigma)$ so we require that the parallel transport maps "depend smoothly" on $\alpha,\sigma$. We also want the parallel transport maps to be behave in the obvious way with respect to reparametrizations of $\alpha$, etc. Using the reparametrization invariance, it is easy to see that our parallel lifts $\tilde{\alpha}_{\sigma}$ which by definition satisfy $\frac{d}{dt} \tilde{\alpha}_{\sigma}|_{t = 0} \in H_{(m,\sigma)}P = H_{\tilde{\alpha}_{\sigma}(0)}P$ in fact satisfy $\frac{d}{dt} \tilde{\alpha}_{\sigma}(t) \in H_{\tilde{\alpha}_{\sigma}(t)}P$ for all $t \in [0,1]$. That is, the tangent vectors of horizontal lifts always belong to the horizontal bundle.
- Going the other way, assume we have a $G$-equivariant subbundle $HP$ of $TP$ such that $VP \oplus HP = TP$ and let $\alpha \colon [0,1] \rightarrow M$ be a smooth curve with $\alpha(0) = m$. Given $\sigma \in P_m$, we want to construct a unique horizontal lift $\tilde{\alpha}_{\sigma} \colon [0,1] \rightarrow P$ of $\alpha$ with $\tilde{\alpha}_{\sigma}(0) = \sigma$ which we will call the parallel transport of $\sigma$ along $\alpha$. More explicitly, we want $\pi \circ \tilde{\alpha}_{\sigma} = \alpha$, $\tilde{\alpha}_{\sigma}(0) = \sigma$ and $\frac{d}{dt} \tilde{\alpha}_{\sigma}(t) \in H_{\tilde{\alpha}_{\sigma}(t)}P$ for all $t \in [0,1]$. Note that for each $t \in [0,1]$ and $\mu \in P_{\alpha(t)}$, there exists a unique horizontal tangent vector in $H_{\alpha(t),\mu}P$ which projects under $d\pi$ to $\dot{\alpha}(t)$. This follows from the direct sum decomposition as $d\pi|_{H_{(\alpha(t),\mu)}P}$ is an isomorphism onto $T_{\alpha(t)}M$ so we just take the the horizontal tangent vector to be $d\pi^{-1}|_{H_{(\alpha(t),\mu)}P}(\dot{\alpha}(t))$. This horizontal vectors glue together to give a well-defined smooth vector field on the total space of $\alpha^{*}(E)$. An integral curve of this vector field starting at $(0,\sigma)$ will give us the required horizontal lift. This integral curve will exist for all $t \in [0,1]$ (and not just for a short time) because of the $G$-equivariance. Finally, using the horizontal lifts we can construct the parallel transport maps and verify that they satisfy the properties used in the previous item.
A Hurewicz connection abstracts and generalizes the first approach to the topological setting. Namely, if $s$ is a Hurewicz connection which provides us with uniquely defined lifts, we can define "parallel transport" by $P_{\alpha,0,t}(\sigma) := s(\sigma, \alpha)(t)$. This parallel transport depends continuously on $\alpha$ and $\sigma$ and that's about it. It doesn't have to provide us with a homeomorphism between the fibers along $\alpha$, it is not necessarily invariant under reparametrizations, etc.
