The icons don't show up
I installed the package, and for some reason many icons don't show up.
Why is that? Thank you.
Seems to be a conflict with another package. What other packages have you installed? You can find out by running the following command in your terminal:
apm list -i
I think it didn't put apm in my PATH, but here's the active packages:

I think it didn't put apm in my PATH
You can install apm by running Install Shell Commands from the application menu.
but here's the active packages:
Disable each of the other packages one-by-one, restarting the editor every time you do. That should help you locate the offending package.
I disabled each of them and both, and got the same result. Disabling file-icons removes the icons, predictably.
Alright, so the only possibility left is that the package wasn't installed correctly. This sometimes happens when a connection is interrupted during an installation (the bundled icon-fonts do contribute to an above-average tarball size…)
Run the steps documented here, and try reinstalling file-icons again. I recommend using NPM instead of APM, since APM (Atom's package manager) seems to lack error-handling whatsoever (therefore resulting in corrupted downloads being "installed"…)
rm -rf ~/.atom/.apm ~/.atom/packages/file-icons
cd ~/.atom/packages
git clone https://github.com/file-icons/atom.git file-icons && cd file-icons
npm install --production # Installs runtime dependencies needed by the package
# Then restart Atom