"Epistemic uncertainty" sounds like a fancy way of saying "ignorance". "Aleatoric" I think means "stochastic" (the cognate of that word in Italian is not terribly uncommon).
Wikipedia says:
...Aleatoric uncertainty, aka statistical uncertainty, which is unknowns that differ each time we run the same experiment. For an example of simulating the take-off of an airplane, even if we could exactly control the wind speeds along the run way, if we let 10 planes of the same make start their trajectories would still differ due to fabrication differences. Similarly, if all we knew is that the average wind speed is the same, letting the same plane start 10 times would still yield different trajectories because we do not know the exact wind
I'm sure most of us are used to just being able to badger him about things in the comments here on LW, but for anyone interested here's the link.