This entire guide is based on an old version of Homebrew/Node and no longer applies. It was only ever intended to fix a specific error message which has since been fixed. I've kept it here for historical purposes, but it should no longer be used. Homebrew maintainers have fixed things and the options mentioned don't exist and won't work.
I still believe it is better to manually install npm separately since having a generic package manager maintain another package manager is a bad idea, but the instructions below don't explain how to do that.
Installing node through Homebrew can cause problems with npm for globally installed packages. To fix it quickly, use the solution below. An explanation is also included at the end of this document.