I want to install ArmCord, however the Snap Store version is broken, so I can't use that. I see that they have a bunch of different packages available on their README page, but I don't know what the differences are or what's compatible with my distro.

I'm also not very familiar with the terminal yet, so I'd appreciate an explanation of what each command necessary for the install does.

As mentioned in ArmCord's GitHub page:

How to run/install it?

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Recommended:

Check releases tab for precompiled packages for Linux, Windows and Mac OS. Alternatively use our Sourceforge mirror.

So you can either download a precompiled binary from Sourceforge, as Archisman Panigrahi already mentioned, or you can go to the Releases section of the project, click Assets on the latest version (3.0.8 currently) and select ArmCord_3.0.8_amd64.deb (click here to directly download it).

After downloading the ArmCord_3.0.8_amd64.deb file, run the following command to install it:

sudo apt install /path/to/ArmCord_3.0.8_amd64.deb

In the above command you need to replace /path/to/ArmCord_3.0.8_amd64.deb with the correct path to your downloaded .deb file (e.g. /home/user/Downloads/ArmCord_3.0.8_amd64.deb if you downloaded it to your Downloads directory and your username is user).

You can then launch ArmCord either from your applications menu or by opening a terminal and running:

armcord

Note: Since you are confused about the packages in Ubuntu, I should mention that Ubuntu uses the .deb package format, that's why the above instructions tell you how to download and install the .deb package.