Compiling is when you convert your human-readable programming language to binary machine code.
makepkg is an Arch-exclusive command that ensures that your apps are built properly in any other Arch-based devices. Usually, all packages are compiled on a powerful machine in some server, and you just have to download the binaries.
But in some cases, they don’t compile apps, or, their process may have delayed, so you can manually compile those apps on your device - that is called building from the source. From what I know, you can’t run makepkg on other Linuces (can we? I’ve never used Arch in depth).
About .deb and .rpm, those are a type of packaging format - in reality, they’re just archives. Think of them as .exe and .msi equivalents. It contains the pre-built app, as well as the repository link, which makes it convenient if you want to upgrade that app in the future.