Well... ("that's a deep subject")... You've apparently got the 16-bit version of AoA. The 32-bit version uses HLA, which is "something completely different" - Masm is bad enough.
You can definitely "read" AoA and "use" Nasm, but it will require some mental "translation" from Masm syntax to Nasm syntax. This isn't particularly difficult, but introduces some "extra" complications and as you know, a beginner doesn't need any "extra" complication!
If you want to learn DOS programming, AoA is a good source of information. This is a fairly "easy" way to start, but DOS is really getting pretty "obsolete" at this point. It still "works" - if your OS supports it or you've got an emulator - but learning the details of DOS may not be the best use of your time. I'm pretty "DOS friendly", and I don't think it's a bad place to start, but I wouldn't spend too much time on it... "Graduate" to 32-bit code as soon as you're "ready" - it's actually easier in a lot of ways, but you'll need to deal with the OS more (DOS will let you "bypass" the OS more easily).
If (when) you get to a specific question about how to make something from AoA work with Nasm (it will, I assure you!), re-reading that section of the Manual may help, or ask here. I "used to have" quite a bit of experience translating Masm to Nasm, but my memory is slip-sliding away...
For 32-bit code - in Nasm syntax - you could look at Dr. Paul Carter's work -
http://www.drpaulcarter.com It's not as "extensive" as AoA, but won't require any "translation". No reason not to use "both"!
We welcome your questions, in any case!
Best,
Frank