AMD64 is the official name of x86_64 because it was created by the company AMD. Intel tried to get the marketplace to move to an incompatible IA64 and failed; the marketplace wanted to run 32bit software on their 64bit processors and went with AMD64.
AMD64 works with Intel or AMD brand cpus (if you bought intel - a tiny fraction went in licensing fees to AMD).
If your cpu is ARM architecture, the NO the AMD64 ISO won't boot, likewise if your box is s390, ppc, ppc64el, IA64 or any other architecture it likewise won't boot.
AMD64 won't boot on i686 cpus, you'll get a message like "This kernel requires an x86-64 CPU, but only detected an i686 CPU".
The linux kernel treats i386, i486, i586 & i686 as upgrades of the same architecture; but Debian (and thus Ubuntu) call all of these architectures just i386, thus any x86 (32-bit) ISO will have i386 in it's name.