What are high-energy electrons?
transported electrons in $\ce{NADH}$ have a higher energy than those in $\ce{FADH2}.$

This is jargon describing the redox potential of the electron carrier $\ce{NADH}/\ce{NAD+}$ vs the electron carrier $\ce{FADH2}/\ce{FAD}.$ So if you compare the two reduction half reactions
$$
\begin{align}
\ce{NAD+ + H+ + 2e- &amp;-&gt; NADH} &amp;\quad E^{\circ '}_\mathrm{red} &amp;= \pu{-0.320 V}\tag{1}\\
\ce{FAD + 2H+ + 2e- &amp;-&gt; FADH2} &amp;\quad E^{\circ '}_\mathrm{red} &amp;= \pu{0.031 V}\tag{2}
\end{align}
$$
$\ce{NADH}$ is the stronger reducing agent.

Could anyone come up with a definition understandable by this non-chemist (i.e. me)?

Another way of saying this is that the reaction of $\ce{NADH}$ with dioxygen is more exergonic (the equilibrium lies further on the side of the products, more free energy is available from it) than the reaction of $\ce{FADH2}$ with dioxygen. For completeness, the reduction half reaction for dioxygen/water is:
$$\ce{1/2 O2 + 2 e- + 2H+ -&gt; H2O} \quad E^{\circ '}_\mathrm{red} = \pu{0.816 V}\tag{3}$$

Do they mean that when these electrons are transferred to the electron carriers they are part of molecular structures in which they sit in high energy positions?

It has to do with how tightly the electrons are bound in one and the other electron carrier. The OP is correct to place the term &quot;high-energy&quot; in quotes when referring to electrons; electrons are indistinguishable, so their energy is with respect to the bound state that they are in.