Yes it uses lightning. I'm not sure how lightning payments on nostr work, presumably it looks up in the nostr profile the lnaddress and then your wallet makes a payment to that address.
Then they call it a zap because, nostr.
Discussion
But the note you can attach to a zap, that also appears in Nostr clients. Is that a 'note' in the same way a post on nostr is a note? Or is that another note that just somehow appears here?
Here is some reading on it:
https://github.com/nostr-protocol/nips/blob/master/57.md
Yes they are kind of nostr note.
I'm pretty sure you can add a message to any lightning network transaction because my wallet of Satoshi would show me what was written in zaps when I was using that to receive zaps