Indeed this is not standard terminology: I made it up when writing the notes. As Bill Dubuque says, it stands for "Euclid's Lemma." I would not have used the term "AP domains" back then, because back then I used the term "irreducible" instead of "atom," even though I knew that the literature on factorization theory went the other way around. (Since then I have changed my mind.)
To be honest, both terms "EL-domain" and "AP-domain" seem pretty bad to me. Let me say a little bit about why things worked out this way. The property that irreducibles are prime is an important one, but in the majority of mathematical usage it is drowned out by the more important property of being a UFD or a factorial domain. If you look at Theorem 15.8 of my notes, you'll see that a domain is a UFD iff all irreducibles are primes and every nonzero nonunit factors as a product of irreducibles ("atomic domain"; I used to write "factorization domain," but I switched that one over eventually). The condition of a domain being atomic is much weaker than it being Noetherian, and the way commutative algebra has developed, Noetherian rings have gotten the lion's share of people's attention.
A good way to think about "EL-domain" is as a meaningful way of keeping the uniqueness of factorizations but not their existence. In this it joins many other classes of rings which are roughly (there is not necessarily a canonical or unique way to do it) "non-Noetherian analogues of more familiar properties that imply Noetherianness (or something like it)." Defining and studying such non-Noetherian analogues is in fact an important theme in contemporary commutative algebra, and it comes up in my notes as well. But the terminology for this is all over the place:
- PID $\mapsto$ Bezout domain
- Dedekind domain $\mapsto$ Prufer domain
- Integrally closed Noetherian domain $\mapsto$ Krull domain
As Serge Lang once said, the terminology should be functorial with respect to the ideas. That is unfortunately not really the case here.